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SCC Bishul, Part II
22.

אָמַר רַב נַחְמָן: בְּחַמָּה — דְּכוּלֵּי עָלְמָא לָא פְּלִיגִי דִּשְׁרֵי. בְּתוֹלְדוֹת הָאוּר — כּוּלֵּי עָלְמָא לָא פְּלִיגִי דַּאֲסִיר. כִּי פְּלִיגִי בְּתוֹלְדוֹת הַחַמָּה: מָר סָבַר גָּזְרִינַן תּוֹלְדוֹת הַחַמָּה אַטּוּ תּוֹלְדוֹת הָאוּר, וּמָר סָבַר לָא גָּזְרִינַן.

Rav Naḥman said: cooking directly in the sun, everyone agrees that one is permitted. Cooking with derivatives of fire, i.e., objects that were heated by fire, everyone agrees that it is prohibited. Where do they argue? With derivatives of the sun, i.e., objects heated with the heat of the sun. This Sage (Chachamim), holds that we issue a decree against derivatives of the sun due to derivatives of fire. And this Sage, (Rabbi Yosei), holds that we do not issue a decree.

דשרי - דאין דרך בישולו בכך וחמה באור לא מיחלפא דליגזר הא אטו הא:

בתולדות האור - כגון אם הוחם הסודר הזה באור מתחילה: דאסיר - דהוי בישול: תולדות חמה - שהוחם הסודר בחמה: אטו תולדות האור - דמאן דחזי סבר דתולדות האור נינהו:

that one is permitted... for this is not the way if cooking, and fire and sunlight are not so interchangeable as to require a decree. with derivatives of fire... e.g. a scarf previously heated by fire. that it is prohibited... for that is bishul. with derivatives of the sun... e.g. a scarf heated in the sun. due to derivatives of fire... for one watching might assume it was heated by fire.

22a.

הַמְבַשֵּׁל הוּא מֵאֲבוֹת מְלָאכוֹת, שֶׁכֵּן בַּמִּשְׁכָּן הָיוּ מְבַשְּׁלִים סַמְמָנִים לִצְבּוֹעַ בָּהּ. וְלֹא בִּשּׁוּל בַּמַּיִם בִּלְבַד, אֶלָּא אַף הַטִּגּוּן וְהָאֲפִיָּה וְהַצְּלִיָּה הֵן בִּכְלַל בִּשּׁוּל.

וְכֵן כָּל מִי שֶׁרִפָּה גּוּף קָשֶׁה בְּאוּר, כְּגוֹן הַמַּתִּיךְ אֶחָד מִמִּינֵי מַתָּכוֹת, אוֹ הַמְחַמֵּם אֶת הַמַּתֶּכֶת עַד שֶׁנַּעֲשָׂה גַּחֶלֶת, וְכֵן הַמְמַסֵּס הַשַּׁעֲוָה אוֹ הַחֵלֶב אוֹ הַזֶּפֶת וְהַגָּפְרִית בְּאוּר, אוֹ שֶׁהִקְשָׁה גּוּף רַךְ בְּאוּר, כְּגוֹן הַמַּנִּיחַ כְּלִי אֲדָמָה בְּאוּר עַד שֶׁנַּעֲשׂוּ כְּלִי חֶרֶס – חַיָּב מִשּׁוּם מְבַשֵּׁל.

וּכְשֵׁם שֶׁאָסוּר לְבַשֵּׁל בְּאוּר – כָּךְ אָסוּר לְבַשֵּׁל בְּתוֹלֶדֶת הָאוּר, כְּגוֹן לִתֵּן בֵּיצָה בְּצַד קְדֵרָה שֶׁהוּחַמָּה בְּאוּר, אוֹ לְשַׁבְּרָהּ עַל הַסּוּדָר שֶׁהוּחַם בְּאוּר כְּדֵי שֶׁתִּצָּלֶה מְעַט. וְאִם נִצְלֵית שָׁם כְּמַאֲכַל בֶּן דְּרוֹסָאִי – חַיָּב מִשּׁוּם מְבַשֵּׁל כְּמוֹ בְּצוֹלֶה עַל הָאוּר.

וַאֲפִלּוּ בְּתוֹלֶדֶת חַמָּה, כְּגוֹן בְּסוּדָר שֶׁהוּחַם בַּחַמָּה  – אָסוּר, גְּזֵרָה מִשּׁוּם תּוֹלְדוֹת הָאוּר, שֶׁהָרוֹאֶה סָבוּר שֶׁהוּחַם הַסּוּדָר בְּאוּר. וְכֵן אָסוּר לְגַלְגְּלָהּ עַל גַּבֵּי חוֹל וַאֲבַק דְּרָכִים שֶׁהוּחַמּוּ מִכֹּחַ חַמָּה. וְאָסוּר לְהַטְמִינָהּ בָּהֶם אֲפִלּוּ מִבְּעוֹד יוֹם, גְּזֵרָה שֶׁמָּא יַטְמִין בְּרֶמֶץ,  כְּמוֹ שֶׁנִּתְבָּאֵר בְּסִימָן רנ"ז.

אֲבָל בַּחַמָּה עַצְמָהּ, כְּגוֹן לִתֵּן בֵּיצָה בַּחַמָּה כְּדֵי שֶׁתִּצָּלֶה, אוֹ לִתֵּן מַיִם בַּחַמָּה כְּדֵי שֶׁיּוּחַמּוּ – מֻתָּר, מִפְּנֵי שֶׁאֵין דֶּרֶךְ בִּשּׁוּל בְּכָךְ. וְאֵין לִגְזוֹר עַל חַמָּה מִשּׁוּם אוּר, מִפְּנֵי שֶׁחַמָּה אֵינָהּ מִתְחַלֶּפֶת בְּאוּר:

Cooking is one of the primary categories of labor [forbidden on Shabbos by Scriptural Law], because [when constructing] the Sanctuary, they would cook herbs to use as dyes. Not only cooking in water, but also frying, baking, and roasting are included in the category of cooking. Similarly, one is liable for cooking if he: a) softens a hard substance in fire, e.g., he melts one of the types of metal, b) heats a [piece of] metal until it chars, c) melts wax, fat, tar, or sulfur using fire, or d) causes a soft substance to become hard through fire, e.g., he places a clay implement in a fire until it becomes an earthenware implement.

Just as it is forbidden to cook with fire, so too is it forbidden to cook with the derivatives of fire, e.g., to place an egg next to a pot that was heated with fire or to break it over a cloth that was heated with fire50 so that it will become slightly roasted. If [the egg] roasts to the extent of maachal ben Derusai’i, he is liable for cooking, as if it roasted over a fire.

It is even forbidden to roast in the derivatives of heat from the sun, e.g., [to roast] on a cloth that was heated by the sun. [This is] a decree [instituted out of concern for] derivatives of fire, for an observer may think that this cloth was heated with fire. Similarly, it is forbidden to roll [an egg] over sand or the dust of the roads that are heated by the sun. Similarly, it is forbidden to insulate [food] in such [sand or dust] even while it is still day [on Friday. This is] a decree lest one insulate [food] in a mixture of ash and hot coals, as explained in sec. 257[:1, 5].

It is, however, permitted to [use the heat of] the sun itself, e.g., to place an egg in the sun so that it will roast or to place water in the sun so that it will be heated, since this is not the ordinary way of cooking. There is no [need to] decree against [using the heat of] the sun lest one use fire, since the sun will not be confused with fire.

23.
Igros Moshe, Orach Chayim vol. 3, ch 52
Thus, in a microwave oven which is as good as an oven for cooking, and those who own such ovens use them more frequently than actual fire ovens, and the reason they are not widespread is because the proliferation hasn't occurred yet, but when the proliferation occurs, certainly the whole world will use them for they are more convenient (lit. they are better). Certainly, (the law regarding microwaves) must be learned out from the cooking by fire that was done in the Mishkan, (and assigned) the level of toldah, which, is just like the principal melacha in all matters of prohibition and penalty.
24.
Shemiras Shabbos Kehilchasa ch. 1 n. 12
Regarding the matter of microwave ovens see Minchas Shlomo (vol.1, ch. 12, subsection 2, under the title גם נראה לענית דעתי, as well as subsection 3 regarding similar machines), that since the current of electricity is not fire, it will be considered as the derivatives of the sun, in which the one who cooks is exempt from the sin offering.
25.
Shulchan Shlomo ch. 318
His honor clarified the matter of a solar water-heater and wrote that it should be investigated whether the water heated in a boiler that is arranged to absorb the rays of the sun, will be considered as heated by the sun or as the derivative of the sun, and concluded that it will be the result of the sun.
But this is puzzling to me. How can it be said that the glass is considered as the product of the sun when the glass does not cook the water, it only concentrates the sun's rays?
26.
Tzitz Eliezer vol. 7 ch. 19
Regarding his honor's comments on the use of water from solar-heaters, it is not that the waters are actually heated by a derivative from the sun, rather they are like waters that were heated by a derivative from the sun, a case which everyone agrees would be permitted, as demonstrated in Shabbos 39a.
,,,
...

The explanation 'his honor' has given on this, because:

This water that has been heated through passing in the collectors which are metal pipes are facing directly into the sun, and the water is being heated not from the pipes, rather directly from the Sun's rays, even though the pipes do heat up a little, (nevertheless), they are not hot enough to heat the water, and the water is being heated by passing through the pipes enough times until it is heated enough from the sun.

These pipes can not be compared to sand or dust of the road which was taught in the Mishna that it is forbidden to use them to roast, because sand or dust of the road have already absorbed much heat from the sun and cook and heat even without the presence of the sun, for if the sand were to be moved into the shade, it's heat would still be sufficient to cook.

It is in that case that it was rabbinically decreed as forbidden so to cook with the secondary power of the sun so that people would not confuse it with cooking from secondary power of fire.

However, if they would heat up only while in the sun,, as in our case, if the pipe would be moved to a different location, it would not have the power to heat at all, and it's metal is not thick enough to heat on it's own, it comes out that the water is being heated directly from the sun.

Even if the pipe were to heat up very much, nevertheless, in my opinion, the water would still be considered heated by power of the sun, this is because thier heat is not from the pipes, but rather directly from the sun's rays. This would be similar to a pot standing on the fire, even though the pot is heated up very much, nevertheless, the water is being heated directly from the fire, this is simple.

Added by: Dovber Chaiton
30.

מַתְנִי׳ כׇּל שֶׁבָּא בְּחַמִּין מֵעֶרֶב שַׁבָּת — שׁוֹרִין אוֹתוֹ בְּחַמִּין בְּשַׁבָּת. וְכׇל שֶׁלֹּא בָּא בְּחַמִּין מֵעֶרֶב שַׁבָּת — מְדִיחִין אוֹתוֹ בְּחַמִּין בַּשַּׁבָּת, חוּץ מִן הַמָּלִיחַ הַיָּשָׁן וְדָגִים מְלוּחִין קְטַנִּים וְקוֹלְיָיס הָאִיסְפְּנִין, שֶׁהַדָּחָתָן זוֹ הִיא גְּמַר מְלַאכְתָּן. גְּמָ׳ כְּגוֹן מַאי? אָמַר רַב סָפְרָא: כְּגוֹן תַּרְנְגוֹלְתָּא דְּרַבִּי אַבָּא.

MISHNA: Any salted food item that was already placed in hot water, i.e., cooked, before Shabbat, one may soak it in hot water even on Shabbat. And anything that was not placed in hot water before Shabbat, one may rinse it in hot water on Shabbat but may not soak it, with the exception of old salted fish and small salted fish and the kolyas ha’ispanin fish, for which rinsing with hot water itself is completion of the prohibited labor of cooking. GEMARA: We learned in the mishna that an item that was cooked before Shabbat may be soaked in hot water on Shabbat. The Gemara asks: In what case would soaking in hot water be required after the item was already cooked? Rav Safra said: In the case of the chicken of Rabbi Abba, which for medical reasons was cooked so thoroughly that it completely dissolved.

מתני' כל שבא בחמין - כלומר שנתבשל: שורין אותו בחמין - כדי שיהא נמוח: וכל שלא בא בחמין - כגון בשר יבש שאוכלין אותו חי ע"י הדחק: מדיחין אותו - ולא אמרי' זהו בישוליה אבל לא שורין: חוץ מן המליח ישן - דג מליח של שנה שעברה: וקולייס האיספנין - שם דג שאוכלין אותו מחמת מלחו ע"י הדחה בחמין והנך אפילו הדחה נמי לא שזהו גמר מלאכתן והוי בשול:

Mishna: Any that was placed in hot water - meaning it was cooked. One may soak it in hot water - So it should melt. And anything that was not placed in hot water - e.g. dry meat that is eaten raw in an emergency. One may rinse it - and we do not say "this is how it is cooked". However, one may not soak it (in hot water). With the exception of old salted fish - salted fish from the previous year. Kolyas ha’ispanin - fish that, because of its saltiness, it is rinsed with hot water before eating. These may not even be rinsed for that would complete their preparation, and that would be considered bishul.

30a.

כָּל דָּבָר לַח יֵשׁ בּוֹ בִּשּׁוּל אַחַר בִּשּׁוּל. דְּהַיְנוּ, תַּבְשִׁיל לַח שֶׁנִּתְבַּשֵּׁל כְּבָר כָּל צָרְכּוֹ וְנִצְטַנֵּן, וַאֲפִלּוּ לֹא נִצְטַנֵּן לְגַמְרֵי אֶלָּא שֶׁאֵין הַיָּד סוֹלֶדֶת בּוֹ, וְחִמְּמוֹ בְּשַׁבָּת עַד שֶׁהַיָּד סוֹלֶדֶת בּוֹ – חַיָּב מִשּׁוּם מְבַשֵּׁל.

אֲבָל אִם הוּא חַם כָּל כָּךְ עַד שֶׁהַיָּד סוֹלֶדֶת בּוֹ וּמַרְתִּיחוֹ יוֹתֵר – אֵין בָּזֶה מִשּׁוּם בִּשּׁוּל. וּמֻתָּר אֲפִלּוּ לְכַתְּחִלָּה לְהַרְתִּיחוֹ סָמוּךְ לָאֵשׁ בְּמָקוֹם שֶׁאֵין לָחוּשׁ לְחִתּוּי, כְּמוֹ שֶׁיִּתְבָּאֵר (וְאִם מֻתָּר לִתְּנוֹ עַל גַּבֵּי כִירָה אוֹ תַּנּוּר – נִתְבָּאֵר בְּסִימָן רנ"ג).

וְיֵשׁ אוֹמְרִים שֶׁאֲפִלּוּ נִצְטַנֵּן לְגַמְרֵי – אֵין בִּשּׁוּל אַחַר בִּשּׁוּל. וּמֻתָּר לְהַרְתִּיחוֹ בְּשַׁבָּת סָמוּךְ לָאֵשׁ, בְּמָקוֹם שֶׁאֵין לָחוּשׁ לְחִתּוּי.

וְנָהֲגוּ לְהָקֵל אִם לֹא נִצְטַנֵּן לְגַמְרֵי, אֶלָּא רָאוּי עֲדַיִן לֶאֱכוֹל מֵחֲמַת חֲמִימוּתוֹ. אֲבָל אִם נִצְטַנֵּן לְגַמְרֵי – נוֹהֲגִין כַּסְבָרָא הָרִאשׁוֹנָה אֲפִלּוּ לְעִנְיַן דִּיעֲבַד, כְּמוֹ שֶׁנִּתְבָּאֵר בְּסִימָן רנ"ג.  וַאֲפִלּוּ לִתְּנוֹ עַל תַּנּוּר בֵּית הַחֹרֶף קֹדֶם שֶׁמַּסִּיקוֹ הַנָּכְרִי, כְּדֵי שֶׁיִּתְחַמֵּם כְּשֶׁיַּסִּיקֶנּוּ – אָסוּר, כְּמוֹ שֶׁנִּתְבָּאֵר שָׁם. וְכֵן בְּכָל דָּבָר שֶׁיִּתְבָּאֵר שֶׁיֵּשׁ בּוֹ מִשּׁוּם בִּשּׁוּל:

[According to Scriptural Law, the forbidden labor of] cooking [also] applies to [reheating] moist food that was fully cooked and then cooled off. Even if [the food] has not entirely cooled, [but has] only [cooled to the point where] it is no longer so hot that a hand would recoil from touching it – [in Talmudic terminology, yad soledes bo] – and one reheats it on Shabbos until it reaches the temperature that a hand would recoil from touching it, he is liable for cooking. If, however, [the food] is so hot that a hand would recoil from touching it, and [one] boils it further, it is not considered cooking. Even as an initial and preferred option, [one] may [place the hot, moist food] close to a fire so that it boils, [provided] there is no concern that he will stoke [the fire], as will be explained. (Whether it is permitted to place [such food] on top of a range or an oven is explained in sec. 253[:14-17]).

There are authorities who maintain that even if the food cools off entirely, [the forbidden labor of] cooking does not apply to food that is already [fully] cooked, and it is permitted to [place the food] close to a fire so that it boils, [provided] there is no concern that he will stoke [the fire].

It is customary to rule leniently [and allow the food to be reheated] as long as [the food] has not cooled entirely, but rather is sufficiently warm that it is fit to be eaten. If, however, [the food] has completely cooled off, it is customary to follow the first opinion, even after the fact, as explained in sec. 253[:25]. It is even forbidden to place [food] on an oven [that heats] a home [in] the winter before a non-Jew kindles it, so that [the food] will warm when [the non-Jew] kindles [the oven], as explained in that source. Similar concepts apply with regard to all activities that are [categorized as] cooking, as will be explained.

31.

תַּבְשִׁיל יָבֵשׁ שֶׁאֵין בּוֹ רֹטֶב כְּלָל – אֵין בּוֹ בִּשּׁוּל אַחַר בִּשּׁוּל, אִם נִתְבַּשֵּׁל כְּבָר כָּל צָרְכּוֹ, וַאֲפִלּוּ נִצְטַנֵּן לְגַמְרֵי. וּמֻתָּר אֲפִלּוּ לִשְׁרוֹתוֹ, בִּכְלִי רִאשׁוֹן רוֹתֵחַ כְּדֵי שֶׁיְּהֵא נִמּוֹחַ שָׁם וְיִהְיֶה דָּבָר לַח.

[The forbidden labor of] cooking does not apply to [reheating] dry food that does not have any liquid on it at all, provided [the food] has already been fully cooked. [This leniency applies] even if [the cooked food] has cooled off entirely. It is even permitted to soak [the food] in a k’li rishon that is boiling, so that it will become soft and moist.

32.
Iggros Moshe vol 4 ch 74; Bishul 7
Response: I don't understand the distinction between "very moist" and "somewhat moist", for even the smallest amount of moisture is subject to cooking when it is cold.
Perhaps it is founded on the principal that "there is no cooking after cooking" with regards to liquid, which makes this situation unclear. It is therefore best to be stringent, other than in a case of great need.
33.
Machatzis Hashekel 318:16
In the case a baked pudding (Kugel) there is no concern of bishul since it is a dry food, and although it will in the end soften and run, still we only consider its present state, and presently it is dry.
33a.
Shulchan Aruch Harav 318:11
(see source 31)
33b.
Siddur Admur Hazaken
Even though there is no "cooking after cooking" with dry foods, still if a portion of it will melt and run it would be subject to the laws of bishul, when it is so hot that the hand would recoil from it. Therefore we must be exceedingly careful not to reheat meat or poultry, roasted or cooked, to the temperature of "yad soledes bo" if there is moisture that will melt and run, because of a prohibition whose penalty is excision or stoning, G-d forbid. Also, we must not pour (hot liquids) from a kli rishon onto sugar, for when the sugar melts it is subject to the laws of bishul.
34.

מַתְנִי׳ הָאִילְפָּס וְהַקְּדֵרָה שֶׁהֶעֱבִירָן מְרוּתָּחִין — לֹא יִתֵּן לְתוֹכָן תַּבְלִין. אֲבָל נוֹתֵן הוּא לְתוֹךְ הַקְּעָרָה אוֹ לְתוֹךְ הַתַּמְחוּי.

MISHNA: In continuation of the discussion of vessels where the prohibition of cooking applies even though the vessels are not actually on the fire itself, the mishna establishes: A stew pot [ilpas] and a pot that were removed from the fire while they were still boiling, even if they were removed before Shabbat, one may not place spices into them on Shabbat itself. Even though the pot is not actually standing on the fire, the spices are still cooked in it because the pot is a primary vessel, i.e., a vessel whose contents were cooked on the fire. However, one may place the spices into a bowl or into a tureen [tamḥui], which is a large bowl into which people pour the contents a stew pot or a pot. Bowls and tureens are both secondary vessels and food placed into them does not get cooked.

34a.

אָסוּר לִתֵּן תְּבָלִין בִּקְעָרָה וְלַעֲרוֹת עֲלֵיהֶם מִכְּלִי רִאשׁוֹן שֶׁהַיָּד סוֹלֶדֶת בּוֹ, מִפְּנֵי שֶׁעֵרוּי מִכְּלִי רִאשׁוֹן מְבַשֵּׁל כְּדֵי קְלִיפָה. וְאִם עֵרָה – אֲזַי הַקְּלִיפָה אֲסוּרָה, מִפְּנֵי שֶׁנִּתְבַּשְּׁלָה בְּשַׁבָּת.

אֲבָל אִם עֵרָה תְּחִלָּה מֵהַכְּלִי רִאשׁוֹן לְתוֹךְ הַקְּעָרָה – מֻתָּר לִתֵּן אַחַר כָּךְ הַתְּבָלִין לַקְּעָרָה,מִפְּנֵי שֶׁחֹם כְּלִי שֵׁנִי אֵין בּוֹ כֹּחַ לְבַשֵּׁל.

וַאֲפִלּוּ לְהַמַּחְמִירִים בִּכְלִי שֵׁנִי לְעִנְיַן אִסּוּר וְהֶתֵּר, שֶׁסּוֹבְרִים שֶׁיֵּשׁ בּוֹ כֹּחַ לְהַבְלִיעַ וּלְהַפְלִיט כָּל זְמַן שֶׁהַיָּד סוֹלֶדֶת בּוֹ, מִכָּל מָקוֹם אֵין בּוֹ כֹּחַ לְבַשֵּׁל אַף עַל פִּי שֶׁהַיָּד סוֹלֶדֶת בּוֹ. וְאֵינוֹ דוֹמֶה לִכְלִי רִאשׁוֹן שֶׁמְּבַשֵּׁל כָּל זְמַן שֶׁהַיָּד סוֹלֶדֶת בּוֹ, לְפִי שֶׁכְּלִי רִאשׁוֹן מִתּוֹךְ שֶׁעוֹמֵד (אֵצֶל הָאֵשׁ ) עַל הָאוּר  – דָּפְנוֹתָיו חַמִּין וּמַחֲזִיק חֻמּוֹ הַרְבֵּה, וּלְכָךְ נָתְנוּ בוֹ שִׁעוּר  כָּל זְמַן שֶׁהַיָּד סוֹלֶדֶת בּוֹ, מַה שֶּׁאֵין כֵּן בִּכְלִי שֵׁנִי שֶׁאֵין דָּפְנוֹתָיו חַמִּין וְהוֹלֵךְ וּמִתְקָרֵר:

It is forbidden to place spices in a bowl and pour [liquid] over them from a k’li rishon from which a hand would recoil, because [hot liquid] poured from a k’li rishon cooks their outer surface. Thus, if one pours [such hot liquid over the spices, their] outer portion becomes forbidden, because it was cooked on Shabbos.

If, by contrast, one first pours [the liquid] from a k’li rishon into the bowl, it is permitted to place spices into the bowl afterwards. [The rationale is that the bowl is a k’li sheni, and] the heat of a k’li sheni is not powerful enough to cook. [True,] there are authorities who rule stringently concerning permitted and forbidden foods, maintaining that a k’li sheni has the power to cause [food to be] absorbed [into the walls of a cooking utensil] or to be emitted [from its walls] as long as it is [so hot that] a hand would recoil [when touching] it. Nevertheless, even [they agree that a k’li sheni] does not have the power to cook despite being [hot enough that] a hand would recoil [when touching] it. [A k’li sheni] is not comparable to a k’li rishon, that can cook as long as it is [so hot that] a hand would recoil [when touching] it. [The reason for the distinction is that] since a k’li rishon stands (near the fire or) over the fire, its walls are hot and they preserve its heat for a significantly longer time. Therefore, [the Sages] established a measure for [a k’li rishon] – [forbidding it] as long as it is [so hot that] a hand would recoil [when touching] it. In contrast, [the contents of] a k’li sheni, whose walls are not [so] hot, will progressively cool.

34b.

וְיֵשׁ מִי שֶׁמִּסְתַּפֵּק וְחוֹשֵׁשׁ מֵחִיּוּב חַטָּאת גַּם בִּכְלִי שֵׁנִי שֶׁהַיָּד סוֹלֶדֶת בּוֹ, מִפְּנֵי שֶׁיֵּשׁ דְּבָרִים רַכִּים שֶׁמִּתְבַּשְּׁלִים גַּם בִּכְלִי שֵׁנִי, וְאֵין אָנוּ בְּקִיאִין בָּהֶן, וְשֶׁמָּא גַּם הַלֶּחֶם נִקְרָא רַךְ לְעִנְיָן זֶה וּמִתְבַּשֵּׁל גַּם בִּכְלִי שֵׁנִי שֶׁהַיָּד סוֹלֶדֶת בּוֹ. וְכֵן יֵשׁ לְהִסְתַּפֵּק סָפֵק זֶה גַּם בִּשְׁאָר דְּבָרִים. וְיֵשׁ לְהַחֲמִיר בְּכֻלָּן, לְבַד מִבִּתְבָלִין, שֶׁהִתִּירוּ חֲכָמִים בְּפֵרוּשׁ לִתְּנָם בִּכְלִי שֵׁנִי, וְכֵן מַיִם וְשֶׁמֶן, כְּמוֹ שֶׁיִּתְבָּאֵר. וְהוּא הַדִּין לִשְׁאָר מַשְׁקִין. וַאֲפִלּוּ בְּבָשָׂר צָלוּי יֵשׁ לְהַחֲמִיר, כֵּיוָן שֶׁיֵּשׁ בִּשּׁוּל אַחַר צְלִיָּה. אֲבָל בִּמְבֻשָּׁל אֵין לְהַחֲמִיר כְּלָל, אֲפִלּוּ בִּכְלִי רִאשׁוֹן, שֶׁלְּדִבְרֵי הַכֹּל אֵין בִּשּׁוּל אַחַר בִּשּׁוּל בְּדָבָר יָבֵשׁ.

אֲבָל יֵשׁ מַתִּירִין בְּאָפוּי וְצָלוּי, אֲפִלּוּ צוֹנְנִים, לִתְּנָם אֲפִלּוּ בְּתוֹךְ כְּלִי רִאשׁוֹן רוֹתֵחַ, מִפְּנֵי שֶׁאֵין בִּשּׁוּל אֲפִלּוּ אַחַר אֲפִיָּה וּצְלִיָּה.

וְנָהֲגוּ לִזָּהֵר לְכַתְּחִלָּה כַּסְבָרָא הָרִאשׁוֹנָה, שֶׁלֹּא לִתֵּן פַּת  אֲפִלּוּ בִּכְלִי שֵׁנִי כָּל זְמַן שֶׁהַיָּד סוֹלֶדֶת בּוֹ. אֲבָל בְּדִיעֲבַד יֵשׁ לְהַתִּיר אֲפִלּוּ בִּכְלִי רִאשׁוֹן כַּסְבָרָא הָאַחֲרוֹנָה:

There is an authority who casts doubts and raises concerns about whether one is liable for a sin-offering [if baked food is placed in] a bowl that is a k’li sheni that a hand would recoil [from when touching] it. [The rationale is that] there are soft foods that will cook in a k’li sheni from which a hand would recoil. We are not knowledgeable regarding [which foods are meant], and it is possible that bread is also considered “soft” in this context and will cook in a k’li sheni from which a hand would recoil.

This question applies equally with regard to other foods, and stringency should be shown with regard to all [foods] except spices, for our Sages specifically allowed them to be placed in a k’li sheni. Similar [laws] apply with regard to water and oil, as will be explained, and also with regard to other liquids.

One should [similarly] be stringent even with regard to roasted meat, because the concept of cooking applies to [food] that has been roasted. There is, however, no need for stringency with regard to cooked [meat], even [when placed] in a k’li rishon, because all authorities agree that [the forbidden labor of] cooking does not apply to food that is dry after it is already [fully] cooked.

There are, however, other authorities who allow placing baked and roasted foods [that are dry], even when they are cold, in a k’li rishon that is boiling, because [they maintain that] the concept of cooking does not apply [to food that has been] baked or roasted. As an initial preference, it is customary to follow the first opinion and be scrupulous not to place bread even in a k’li sheni as long as a hand would recoil [when touching] it. After the fact, however, even [bread placed] in a k’li rishon is permitted, as the second opinion [maintains].

35.
Shmiras Shabbos Kehilchasa
Since there are foods that will be cooked in a kli sheini, and we are not fluent in knowing which ones will, this is cause to be stringetn and to sy that any food that is not cooked is subject to the prohibition of bishul even when placed in a kli sheini, provided the food in the kli sheini is yad soledes bo.

Therefore, it is forbidden to put a raw egg into a hot food on your plate. The same rule applies to regular cocoa, ground coffee beans that have not been cooked, tea leaves in any form i.e. tea bags or tea cup, honey, mint leaves etc. All these should not be put into a cup that is kli sheini if they have not been cooked beforehand. Similarly, a lemon slice should not be put into a tea that is kli sheini.
The exceptions to the rule are: water, oil and spices. These are allowed to be placed, uncooked, into a kli sheini.
nn. And what counts as spices? I heard from RSZA zt"l that the spices that are found among us are not in that category, since they are ground very finely and they will certainly cook in a kli sheini.

35a.

וְאִם הוּא דָּבָר יָבֵשׁ שֶׁלֹּא נִתְבַּשֵּׁל מִלִּפְנֵי הַשַּׁבָּת – אֵין שׁוֹרִין אוֹתוֹ בְּשַׁבָּת בְּחַמִּין שֶׁהַיָּד סוֹלֶדֶת בּוֹ, אֲפִלּוּ אֵינוֹ נִמּוֹחַ שָׁם כְּלָל. וְאִם שָׁרָה בִּכְלִי רִאשׁוֹן – חַיָּב.

וּבִכְלִי שֵׁנִי אָסוּר לִשְׁרוֹתוֹ מִדִּבְרֵי סוֹפְרִים, מִפְּנֵי שֶׁנִּרְאֶה כִּמְבַשֵּׁל. וְאֵינוֹ דוֹמֶה לִתְבָלִין שֶׁמֻּתָּר לִתְּנָם בִּכְלִי שֵׁנִי כְּמוֹ שֶׁיִּתְבָּאֵר, מִפְּנֵי שֶׁהַתְּבָלִין עֲשׂוּיִין לְמַתֵּק הַקְּדֵרָה, וְאֵינוֹ נִרְאֶה כִּמְבַשֵּׁל. וְכֵן כָּל דָּבָר הֶעָשׂוּי לִתֵּן טַעַם בִּקְדֵרָה, כְּגוֹן שׁוּם וּבָצָל – הֲרֵי הֵם כִּתְבָלִין, וּמֻתָּר לִתְּנָם בִּכְלִי שֵׁנִי אַף עַל פִּי שֶׁהַיָּד סוֹלֶדֶת בּוֹ. וְיֵשׁ מִי שֶׁמִּסְתַּפֵּק בְּבָצָל, שֶׁמָּא הוּא מִתְבַּשֵּׁל גַּם בִּכְלִי שֵׁנִי כָּל זְמַן שֶׁהַיָּד סוֹלֶדֶת בּוֹ. וְנָהֲגוּ לְהָקֵל. וְיֵשׁ מַתִּירִין בְּכָל דָּבָר יָבֵשׁ שֶׁלֹּא נִתְבַּשֵּׁל לִשְׁרוֹתוֹ בִּכְלִי שֵׁנִי שֶׁהַיָּד סוֹלֶדֶת בּוֹ. וְיֵשׁ לְהַחֲמִיר בָּזֶה כַּסְבָרָא הָרִאשׁוֹנָה.

Dry food that was not cooked at all before Shabbos may not be soaked on Shabbos [in water]62 so hot that it will cause a hand to recoil, even if it does not soften there at all. If one soaks [dry food] in a k’li rishon, he is liable [for the forbidden labor of cooking].

Soaking [dry food] in a k’li sheni is forbidden according to Rabbinic Law, because it appears like [the person] is cooking. [Dry food] is not comparable to spices, which are permitted to be placed in a k’li sheni, as will be explained. [The rationale is that] spices are added to flavor the cooked food. Hence, [adding them] does not appear the same as cooking. Similarly, all substances that are added to flavor a cooked food, e.g., garlic or onion, are considered as spices and may be added [to food] in a k’li sheni,even if [the food is so hot that] a hand would recoil [when touching] it. There is an authority who raises a question concerning an onion, for perhaps it also cooks in a k’li sheni as long as a hand would recoil [when touching] it. It is, however, customary to rule leniently. There are authorities who permit soaking any dry substance that has not been cooked in a k’li sheni from which a hand would recoil. One should, however, rule stringently in this matter, following the first opinion.

36.
Yalkut Yosef 318
46. Kli sheini doesn't cook even when the hand would recoil from it. [Kli sheini means the dish that you poured the food into from the pot that on the fire]. Therefore, after the food has been poured into it from the kli rishon, it is permissible to place any food that has not been cooked into it to warm [with the exception of those things that cook swiftly]. Similarly it is permissible to pour even a small amount of cold water into a large amount of hot water in a kli sheini even when the hand would recoil.
49. Many have the custom of putting mint into hot tea on Shabbos, even when the tea is so hot that the hand would recoil from it. They have authorities upon whom they can rely who rule that kli sheini doesn't cook, and and mint leaves are not among the kalei habishul that would cook in a kli sheini. Those who are stringent should be blessed. But we must be careful that the leaves be free of worms.
37.
Chazon Ish 52:18-19
The Mishna Berurah writes, quoting the Magen Avraham that we must be stringent regarding all things that the gemara has not explicitly permitted. However, it could be said that since oil and water have been expressly described in the Talmud as not cooking swiftly, there should be no stringency with regard to anything other than bread, as described by the SMAG, due to its being baked, which primes it for swift cooking, but not other things.
And it appears that it would be permissible to put lemon in a kli sheini that the hand would recoil from, for since it appears as a cooked dish, there is no concern, for it looks like spices...

וְיֵשׁ לְהַחֲמִיר בְּכֻלָּן, לְבַד מִבִּתְבָלִין, שֶׁהִתִּירוּ חֲכָמִים בְּפֵרוּשׁ לִתְּנָם בִּכְלִי שֵׁנִי, וְכֵן מַיִם וְשֶׁמֶן, כְּמוֹ שֶׁיִּתְבָּאֵר. וְהוּא הַדִּין לִשְׁאָר מַשְׁקִין.

... stringency should be shown with regard to all [foods] except spices, for our Sages specifically allowed them to be placed in a k’li sheni. Similar [laws] apply with regard to water and oil, as will be explained, and also with regard to other liquids...

38.
Shmiras Shabbos Kehilchasa 1
64. Dry things that come in lumps, such as potatoes, a piece of meat or whole beans, which are so hot that the hand recoils from them, even after being put into a second dish should be treated as a first dish, because they store the heat.
n. I heard from RSZA taht since he is not intending to cook, and it matters not to him, moreover, the cucumber is somewhat ruined by this and he is doing it in a manner of mis'asek, moreover, the law of gush (lumpy foods) is only a safek, especially in kli shlishi, therefore there is no reason to be concerned about this. Especially when the cucumbers were pickled which is akin to cooking.
39.
Shmiras Shabbos Kehilchasa 1, n. 200
Ketchup is not really a liquid and its status is that of a cooked tomato, which although it has juice it is treated as a dry food. Therefore it is more lenient.
40.
Iggros Moshe vol 4 72; Bishul 5
And this that you are in doubt regarding Ketchup, (That) perhaps it is considered a food because it is thick, even though it is poured.
It seems that this does not depend on whether it is considered a food or liquid in regards to the laws of impurity.
Rather (it depends on), if the item flows, or if it is a solid piece that does not need a container to keep it together- in which case it would be considered a liquid.
Practically, it seems like the opinion of the , that it is permitted to put ketchup on a piece of hot meat that was put into a kli sheini.
For even if we were to be stringent and keep both stringencies, these two stringencies are both stringencies that we are doubtful if we have to keep them on Shabbos.
Over here though,1) the stringency of "there is cooking after cooking in regards to a liquid" is an 'outside' stringency, for the great Posskim hold that there is "no cooking after cooking in regards to a liquid". The Rem"a agrees with this as well, and only in a case that it's completely cooled are we stringent.
2) The stringency of considering a solid food on a kli sheini as being hot, many argue on this, including the Rem"a who is a (most) fundamental opinion.
40a.

אָסוּר לָטוּחַ שֶׁמֶן וְשׁוּם עַל גַּבֵּי הַצָּלִי בְּעוֹדוֹ חַם שֶׁהַיָּד סוֹלֶדֶת בּוֹ, אֲפִלּוּ נִצְלָה מִבְּעוֹד יוֹם, דְּמִכָּל מָקוֹם יִתְבַּשֵּׁל הַשּׁוּם וְהַשֶּׁמֶן. אֲבָל שֻׁמָּן שֶׁנִּקְרַשׁ – אֵין בּוֹ מִשּׁוּם בִּשּׁוּל, אֶלָּא שֶׁיֵּשׁ לְהִסְתַּפֵּק בּוֹ מִצַּד אַחֵר, כְּמוֹ שֶׁנִּתְבָּאֵר לְמַעְלָה:

It is forbidden to baste roasted meat190 with [uncooked] oil and garlic while [the meat] is still so hot that a hand would recoil [when touching] it. [This stringency applies] even if the meat was roasted while it was still day on Friday, for the garlic and the oil will nevertheless cook [on Shabbos. The prohibition against] cooking does not apply, however, to congealed fat.191 Nevertheless, there is uncertainty regarding warming [the fat] for another reason, as explained above.

190. The commentaries have questioned whether this ruling applies to cooked food that was removed from a k’li rishon and placed in a k’li sheni without sauce.
In his Kuntreis Acharon (note 11)to sec. 253, the Alter Rebbe writes that the leniencies of a k’li sheni do not apply with regard to solid food. The rationale is that the reason given for the leniency of a k’li sheni is that the walls of the utensil are cold and they cool the food, as mentioned in subsection 19 above. Liquid placed in a utensil spreads out and touches the utensil’s walls, in contrast to solid food that does not spread out and touch the walls of the utensil. Thus, the walls have no cooling effect on the solid food, and therefore, the leniency of a k’li sheni does not apply. See also subsection 20 above, which mentions placing a solid hot piece of meat in cold sauce. See Tzemach Tzedek, Yoreh Deah, responsum 65, which also discusses this issue.

191. See subsection 27.
192. It is considered as if one brought the fat into being on Shabbos (nolad), as stated in subsection 25. See also subsection 28, which discusses heating solid fat.
41.
Sefer Yireim pg. 134
Therefore, a person should be careful on Shabbos not to put anything into a kli sheini, or even into a kli shlishi which is "hot enough to make the hand recoil".
For we are not expert to determine which items are 'soft' or 'hard' - which items would cook in a kli sheini and those that would not be cooked.
42.
Iggros Moshe vol 4; Bishul 15
From my perspective there is no reason to say that anything cooks in the third dish, for we have not found (from our predecessors) only that there are things that cook in a second dish, and since we are unsure which items they are, we must be stringent about everything. We have not found (sources that say) that things cook in the third dish.
With regards to what is written in ??? 318:28 that we see with our eyes that tea cooks, it is a misunderstanding. What our eyes see is that the water is colored, but that is not a function of cooking, for even cold water would eventually become colored by tea leaves.
43.
Mishne Berurah "Dirshu" 318 n. 51
Ginas Vradim writes regarding coffee, that since it is the main focus of the drink [not like spices which only add flavor] it is forbidden to put it into a kli sheini because it "looks like cooking".
Ohr Zaruah implies that even putting spices into kli sheini is problematic, and is only allowed because sometimes even cold dishes are spiced.
44.
Mishne Berurah "Dirshu" 318 n. 56
Pasteurized milk, the Minchas Yitzchak and the Shevet Halevi write that it can be considered "previously cooked" and can be poured into kli sheini. RSZA and RMF imply the same. On the other hand, RYS Elyashiv holds that pasteurization isn't considered cooking in our context. And since the cooking of the milk was not completed, there is a real possibility of "appearing like cooking" when milk is poured into kli sheini. In fact, he adds, that in his home they were scrupulous to boil the milk before Shabbos.
45.

יוֹסֵי אוֹמֵר: יוֹצְאִין בְּרָקִיק הַשָּׁרוּי, אֲבָל לֹא בִּמְבוּשָּׁל....

Rabbi Yosei says: One can fulfill his obligation with a wafer that has been soaked in a cooked dish but not with a boiled wafer...

אבל לא במבושל - דלאו לחם קרינא ביה ואע"פ שכבר נאפה בתנור הואיל וחזר ובישלו ביטלו מתורת לחם:

46.
Yalkut Yosef 318: Bishul Achar Afiya Ukliya, 61
...
46a.

יֵשׁ מִי שֶׁאוֹמֵר,קמ שֶׁאַף עַל פִּי שֶׁאֵין בִּשּׁוּל אַחַר בִּשּׁוּל בְּדָבָר יָבֵשׁ,83 מִכָּל מָקוֹם יֵשׁ בִּשּׁוּל אַחַר אֲפִיָּה אוֹ צְלִיָּה, שֶׁהָאָפוּי אוֹ הַצָּלוּי, אִם נְתָנוֹ, אֲפִלּוּ בְּעוֹדוֹ רוֹתֵחַ, בְּתוֹךְ כְּלִי רִאשׁוֹן שֶׁהַיָּד סוֹלֶדֶת בּוֹקמא – חַיָּב.קמב

וּלְפִי זֶה גַּם בִּכְלִי שֵׁנִי יֵשׁ לְהַחֲמִיר מִדִּבְרֵי סוֹפְרִים,קמג, 84 כְּמוֹ שֶׁנִּתְבָּאֵר לְמַעְלָה.קמד וְלָכֵן צָרִיךְ לִזָּהֵר שֶׁלֹּא לִתֵּן פַּת אֲפוּיָה אֲפִלּוּ בִּקְעָרָה שֶׁהִיא כְּלִי שֵׁנִי, כָּל זְמַן שֶׁהַיָּד סוֹלֶדֶת בּוֹ.85

וְיֵשׁ מִי שֶׁמִּסְתַּפֵּקקמה וְחוֹשֵׁשׁ מֵחִיּוּב חַטָּאת גַּם בִּכְלִי שֵׁנִי שֶׁהַיָּד סוֹלֶדֶת בּוֹ, מִפְּנֵי שֶׁיֵּשׁ דְּבָרִים רַכִּים שֶׁמִּתְבַּשְּׁלִים גַּם בִּכְלִי שֵׁנִי, וְאֵין אָנוּ בְּקִיאִין בָּהֶן, וְשֶׁמָּא גַּם הַלֶּחֶםקמו נִקְרָא רַךְ לְעִנְיָן זֶה וּמִתְבַּשֵּׁל גַּם בִּכְלִי שֵׁנִי שֶׁהַיָּד סוֹלֶדֶת בּוֹ. וְכֵן יֵשׁ לְהִסְתַּפֵּק סָפֵק זֶה גַּם בִּשְׁאָר דְּבָרִים. וְיֵשׁ לְהַחֲמִיר בְּכֻלָּן,קמז לְבַד מִבִּתְבָלִין, שֶׁהִתִּירוּ חֲכָמִים בְּפֵרוּשׁ לִתְּנָם בִּכְלִי שֵׁנִי,קמח, 86 וְכֵן מַיִם וְשֶׁמֶן, כְּמוֹ שֶׁיִּתְבָּאֵר.קמט, 87 וְהוּא הַדִּין לִשְׁאָר מַשְׁקִין. וַאֲפִלּוּ בְּבָשָׂר צָלוּי יֵשׁ לְהַחֲמִיר,קנ כֵּיוָן שֶׁיֵּשׁ בִּשּׁוּל אַחַר צְלִיָּה. אֲבָל בִּמְבֻשָּׁל אֵין לְהַחֲמִיר כְּלָל, אֲפִלּוּ בִּכְלִי רִאשׁוֹן, שֶׁלְּדִבְרֵי הַכֹּל אֵין בִּשּׁוּל אַחַר בִּשּׁוּל בְּדָבָר יָבֵשׁ.קנא

אֲבָל יֵשׁ מַתִּירִיןקנב בְּאָפוּי וְצָלוּי,88 אֲפִלּוּ צוֹנְנִים, לִתְּנָם אֲפִלּוּ בְּתוֹךְ כְּלִי רִאשׁוֹן רוֹתֵחַ,89 מִפְּנֵי שֶׁאֵין בִּשּׁוּל אֲפִלּוּ אַחַר אֲפִיָּה וּצְלִיָּה.קנג

וְנָהֲגוּ לִזָּהֵר לְכַתְּחִלָּה כַּסְבָרָא הָרִאשׁוֹנָה, שֶׁלֹּא לִתֵּן פַּת 90 אֲפִלּוּ בִּכְלִי שֵׁנִי כָּל זְמַן שֶׁהַיָּד סוֹלֶדֶת בּוֹ.קנד אֲבָל בְּדִיעֲבַד יֵשׁ לְהַתִּיר אֲפִלּוּ בִּכְלִי רִאשׁוֹן כַּסְבָרָא הָאַחֲרוֹנָה:

יג.

וּלְפִי סְבָרָא הָרִאשׁוֹנָה שֶׁיֵּשׁ בִּשּׁוּל אַחַר אֲפִיָּה וּצְלִיָּה – יֵשׁ גַּם כֵּן אֲפִיָּה וּצְלִיָּה אַחַר בִּשּׁוּל,קנה דְּהַיְנוּ שֶׁכָּל דָּבָר הַמְבֻשָּׁל אָסוּר לִתְּנוֹקנו בְּלֹא רֹטֶבקנז סָמוּךְ לָאֵשׁ, בְּמָקוֹם שֶׁיָּכוֹל לְהִתְחַמֵּם שָׁם עַד שֶׁתְּהֵא הַיָּד סוֹלֶדֶת בּוֹ.קנח

אֲבָל לְדִבְרֵי הַכֹּל אֵין אֲפִיָּה וּצְלִיָּה אַחַר אֲפִיָּה וּצְלִיָּה,קנט וְדָבָר הָאָפוּי אוֹ צָלוּי מֻתָּר לִתְּנוֹ סָמוּךְ לָאֵשׁ92 עַל דֶּרֶךְ שֶׁיִּתְבָּאֵר,קס, 93 אֲפִלּוּ נִצְטַנֵּן כְּבָר לְגַמְרֵי:קסא

There is an authority who maintains that although [the forbidden labor of] cooking does not apply to [reheating] dry food that has already been [fully] cooked [and cooled off,83 heating food that] has [previously] been baked or roasted [is considered as cooking]. Thus, if one placed [food that] was baked or roasted – even if it was still boiling hot – into a k’li rishon from which a hand would recoil, he is liable. Based on this [opinion], according to Rabbinic Law, one should also be stringent with regard to a k’li sheni, as explained above.84 Therefore, one should be careful not to place baked bread even in a bowl that is a k’li sheni as long as a hand would recoil [when touching] it.85

There is an authority who casts doubts and raises concerns about whether one is liable for a sin-offering [if baked food is placed in] a bowl that is a k’li sheni that a hand would recoil [from when touching] it. [The rationale is that] there are soft foods that will cook in a k’li sheni from which a hand would recoil. We are not knowledgeable regarding [which foods are meant], and it is possible that bread is also considered “soft” in this context and will cook in a k’li sheni from which a hand would recoil.

This question applies equally with regard to other foods, and stringency should be shown with regard to all [foods] except spices, for our Sages specifically allowed them to be placed in a k’li sheni.86 Similar [laws] apply with regard to water and oil, as will be explained,87 and also with regard to other liquids.

One should [similarly] be stringent even with regard to roasted meat, because the concept of cooking applies to [food] that has been roasted. There is, however, no need for stringency with regard to cooked [meat], even [when placed] in a k’li rishon, because all authorities agree that [the forbidden labor of] cooking does not apply to food that is dry after it is already [fully] cooked.

There are, however, other authorities who allow placing baked and roasted foods88 [that are dry], even when they are cold, in a k’li rishon that is boiling,89 because [they maintain that] the concept of cooking does not apply [to food that has been] baked or roasted. As an initial preference, it is customary to follow the first opinion and be scrupulous not to place bread90 even in a k’li sheni as long as a hand would recoil [when touching] it. After the fact, however, even [bread placed] in a k’li rishon is permitted, as the second opinion [maintains].91