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The Mishnah on 73a lists the 39 melachos. Number 11 is one who bakes.
האופה - לא הוה במשכן דלא שייך אלא בפת ופת לא שייכא במלאכת המשכן אבל כולהו קמייתא הואי בסממנין של צבע תכלת וארגמן ותולעת שני ובגמרא פריך דשביק תנא מבשל דהוי בסממנים ונקט אופה ושיעורן של אלו כגרוגרת חוץ מחורש דבכל שהוא לקמן בפרק הבונה (שבת דף קג.):

Haofeh - [to be an Av it needs to have been in the Creation of the Mishkan. So where was it?] It was not in the creation of the mishkan since it can only be done to make bread, and bread was not used to make the mishkan. Though the whole set of melachos 1-11 all were there [making them Avos] in the preparation of the dyes: blue, crimson, and scarlet. In the gemara we ask, why did the Tanna leave out the word cooking and use baking instead?...

וְהַלָּשׁ וְהָאוֹפֶה. אָמַר רַב פָּפָּא: שְׁבַק תַּנָּא דִידַן בִּישּׁוּל סַמְמָנִין דַּהֲוָה בְּמִשְׁכָּן וּנְקַט אוֹפֶה?! תַּנָּא דִידַן סִידּוּרָא דְפַת נָקֵט.
We learned in the mishna, among those liable for performing primary categories of labor: And one who kneads and one who bakes. Rav Pappa said: Our tanna left out the labor of cooking the spices for dye, which was performed in the Tabernacle, and included the labor of baking, which was not performed in the construction of the Tabernacle. If, as stated above, all the primary categories of labor were derived from the labors in the Tabernacle, why did the tanna omit cooking? The Gemara answers: Our tanna cited the sequence of preparing bread, which was the underlying principle behind his organization of the primary categories of labor. He opened with plowing and concluded with the preparation of bread.
Rabbi Berkowitz says that the logic to doing so is that it was easier to remember the process in terms of making bread than dying, which presumably less people are familiar with. [Source?]

(א) הָאוֹפֶה כִּגְרוֹגֶרֶת חַיָּב. אֶחָד הָאוֹפֶה אֶת הַפַּת אוֹ הַמְבַשֵּׁל אֶת הַמַּאֲכָל אוֹ אֶת הַסַּמְמָנִין אוֹ הַמְחַמֵּם אֶת הַמַּיִם הַכּל עִנְיָן אֶחָד הוּא. שִׁעוּר הַמְחַמֵּם אֶת הַמַּיִם כְּדֵי לִרְחֹץ בָּהֶן אֵיבָר קָטָן. וְשִׁעוּר מְבַשֵּׁל סַמְמָנִין כְּדֵי שֶׁיִּהְיוּ רְאוּיִין לְדָבָר שֶׁמְּבַשְּׁלִין אוֹתָן לוֹ:

(1) One who bakes [something] the size of a dried fig is liable. It is the same whether he bakes bread, cooks a food or herbs or heats up water – it all has one function. The [requisite] amount of heating water is enough to bathe a small limb. And the [requisite] amount of cooking herbs is enough that they be fit for the purpose for which they are being cooked.

Why does the Rambam single out heating up water, more than any other food or drink? [Why isn't it just included in cooking food?]

(ג) הַמַּפְקִיעַ אֶת הַבֵּיצָה בְּבֶגֶד חַם אוֹ בְּחוֹל וּבַאֲבַק דְּרָכִים שֶׁהֵן חַמִּים מִפְּנֵי הַשֶּׁמֶשׁ אַף עַל פִּי שֶׁנִּצְלֵית פָּטוּר. שֶׁתּוֹלְדוֹת חַמָּה אֵינָם כְּתוֹלְדוֹת הָאֵשׁ. אֲבָל גָּזְרוּ עֲלֵיהֶן מִפְּנֵי תּוֹלְדוֹת הָאוּר. וְכֵן הַמְבַשֵּׁל בְּחַמֵּי טְבֶרְיָה וְכַיּוֹצֵא בָּהֶם פָּטוּר. הַמְבַשֵּׁל עַל הָאוּר דָּבָר שֶׁהָיָה מְבֻשָּׁל כָּל צָרְכּוֹ אוֹ דָּבָר שֶׁאֵינוֹ צָרִיךְ בִּשּׁוּל כְּלָל פָּטוּר:

(3) One who cracks an egg on a hot garment or sand or dirt of the roads which are hot because of the heat of the sun is exempt, even though it gets roasted. For the effects of the sun are not like the effects of fire [regarding that which is defined as cooking]. But they decreed about them [to forbid them] on account of the effects of fire. Likewise one who cooks with the hot springs of Tiberius or that which is similar to them is exempt. One who cooks something that is totally cooked or something that does not need to be cooked at all on the fire is exempt.

Remember, פטור means on a Torah level, but it is still forbidden Rabbinically. [As opposed to Mutar.]
Based on our description of the Avos Melachos as something which changes an item, from the end of this piece in the Rambam, we see that the Av of cooking is only when the cooking changes the food into a more edible state. But if it does not need cooking, either because it has been fully cooked or because it did not need cooking at all, putting it on the fire would not be the Torah prohibition of Bishul. [Although it seems that it would still be rabbinically forbidden.]
[I assume we'll see later what the Rambam's "doesn't need cooking at all" refers to: something which is edible raw but can be improved with cooking? Or something which whether or not cooking may have helped it in the past, in it's current state, cooking will not improve it?]
תָּנוּ רַבָּנַן: מֵבִיא אָדָם קִיתוֹן מַיִם וּמַנִּיחוֹ כְּנֶגֶד הַמְּדוּרָה. לֹא בִּשְׁבִיל שֶׁיֵּחַמּוּ, אֶלָּא בִּשְׁבִיל שֶׁתָּפוּג צִינָּתָן. רַבִּי יְהוּדָה אוֹמֵר: מְבִיאָה אִשָּׁה פַּךְ שֶׁל שֶׁמֶן וּמְנִיחָתוֹ כְּנֶגֶד הַמְּדוּרָה. לֹא בִּשְׁבִיל שֶׁיִּבְשַׁל, אֶלָּא בִּשְׁבִיל שֶׁיִּפְשַׁר. רַבָּן שִׁמְעוֹן בֶּן גַּמְלִיאֵל אוֹמֵר: אִשָּׁה סָכָה יָדָהּ שֶׁמֶן וּמְחַמַּמְתָּהּ כְּנֶגֶד הַמְּדוּרָה, וְסָכָה לִבְנָהּ קָטָן, וְאֵינָהּ חוֹשֶׁשֶׁת.
Similarly, the Sages taught: One may bring a jug [kiton] full of cold water and place it opposite the bonfire on Shabbat; not so that the water will heat up, as it is prohibited to cook on Shabbat, rather to temper the cold, as one is permitted to render water less cold on Shabbat. Rabbi Yehuda says: A woman may take a cruse of oil and place it opposite the bonfire; not so the oil will cook, rather, so it will warm until it is lukewarm. Rabban Shimon ben Gamliel says: A woman may smear her hand with oil, and heat it opposite the fire, and afterward smear her young son with the heated oil, and she need not be concerned about cooking on Shabbat.
There are 3 opinions in this Braisa:
1: Tanna Kamma holds that water can be placed near the fire to reduce it's chill.
2: Rabbi Yehuda holds the same about oil.
3:Raban Shimon Ben Gamliel holds that oil not in a container, rather on one's body, may be heated inadvertently.
Keeping the concept of Bishul in mind: what change is there to the water if it is heated up? What change is there to the oil? How does this fit with the Rambam, that an item which is edible without heating is not Assur M'Doraysa to "cook"?
אִיבַּעְיָא לְהוּ: שֶׁמֶן מָה הוּא לְתַנָּא קַמָּא? רַבָּה וְרַב יוֹסֵף דְּאָמְרִי תַּרְוַיְיהוּ לְהֶתֵּירָא, רַב נַחְמָן בַּר יִצְחָק אָמַר לְאִיסּוּרָא. רַבָּה וְרַב יוֹסֵף דְּאָמְרִי תַּרְוַיְיהוּ לְהֶתֵּירָא: שֶׁמֶן אַף עַל פִּי שֶׁהַיָּד סוֹלֶדֶת בּוֹ — מוּתָּר. קָסָבַר תַּנָּא קַמָּא שֶׁמֶן אֵין בּוֹ מִשּׁוּם בִּשּׁוּל. וַאֲתָא רַבִּי יְהוּדָה לְמֵימַר שֶׁמֶן יֵשׁ בּוֹ מִשּׁוּם בִּשּׁוּל, וְהֶפְשֵׁרוֹ לֹא זֶה הוּא בִּשּׁוּלוֹ. וַאֲתָא רַבָּן שִׁמְעוֹן בֶּן גַּמְלִיאֵל לְמֵימַר שֶׁמֶן יֵשׁ בּוֹ מִשּׁוּם בִּשּׁוּל, וְהֶפְשֵׁרוֹ זֶהוּ בִּשּׁוּלוֹ.
A dilemma was raised before the Sages: With regard to heating oil in this manner on Shabbat, what is its legal status according to the first tanna, who permits doing so with water? Does he permit oil as well? Rabba and Rav Yosef both said that the opinion of the first tanna is to permit doing so in the case of oil. Rav Naḥman bar Yitzḥak said that the opinion of the first tanna is to prohibit doing so. Rabba and Rav Yosef both said that the opinion of the first tanna is to permit doing so. The Gemara explains the dispute in the mishna: Oil, even though it is heated to the point at which the hand spontaneously recoils [soledet] from its heat, is permitted to be heated in this manner. The reason is because the first tanna holds that oil is not subject to the prohibition of cooking. Cooking oil to its boiling point requires a very high temperature; merely heating it is not considered cooking. And Rabbi Yehuda came to say that oil is subject to the prohibition of cooking; however, warming it to a lukewarm temperature is not tantamount to cooking it. Therefore, it is permitted to place a jar of oil near the fire in order to raise its temperature, though it is prohibited to heat it to the point of cooking. And Rabban Shimon ben Gamliel came to say that oil is subject to the prohibition of cooking, and warming it is tantamount to cooking it. He permitted it only in the specific case of a woman who smeared her hand with oil, heated it, and smeared her son with it.
It seems we pasken like Rabbi Yehuda. [The Tanna, and like the Amora who quoted Shmuel saying essentially like him.]
To explain the different views, Rabbi Berkowitz frames the dispute in the question of "what is the purpose for which I am using the oil?" According to the first opinion, the use is eating. Therefore, whether the oil is cold or hot it does not become more edible. [Based on gemaras elsewhere, I'm not sure how edible it is anyways?] While according to Rabbi Yehuda, the purpose is to anoint oneself with it. Cold oil doesn't work as well as hot oil. What about warm oil? That would be the dispute between Rabbi Yehuda and Raban Shimon ben Gamliel: Rabbi Yehuda holds that hot oil can be anointed, warm oil is not good enough. While Raban Shimon ben Gamliel holds that the change to warm oil is already "making it useable". [Based on the end where we say that the difference between warm and hot is boiling, how useable is boiling oil for anointing? Or eating?]

רַב נַחְמָן בַּר יִצְחָק אָמַר לְאִיסּוּרָא: שֶׁמֶן אַף עַל פִּי שֶׁאֵין הַיָּד סוֹלֶדֶת בּוֹ — אָסוּר. קָסָבַר שֶׁמֶן יֵשׁ בּוֹ מִשּׁוּם בִּשּׁוּל, וְהֶפְשֵׁרוֹ זֶהוּ בִּשּׁוּלוֹ. וַאֲתָא רַבִּי יְהוּדָה לְמֵימַר הֶפְשֵׁרוֹ לֹא זֶהוּ בִּשּׁוּלוֹ. וַאֲתָא רַבָּן שִׁמְעוֹן בֶּן גַּמְלִיאֵל לְמֵימַר שֶׁמֶן יֵשׁ בּוֹ מִשּׁוּם בִּשּׁוּל, וְהֶפְשֵׁרוֹ זֶהוּ בִּשּׁוּלוֹ. רַבָּן שִׁמְעוֹן בֶּן גַּמְלִיאֵל הַיְינוּ תַּנָּא קַמָּא? אִיכָּא בֵּינַיְיהוּ כִּלְאַחַר יָד. אָמַר רַבִּי יְהוּדָה אָמַר שְׁמוּאֵל: אֶחָד שֶׁמֶן וְאֶחָד מַיִם, יָד סוֹלֶדֶת בּוֹ — אָסוּר, אֵין יָד סוֹלֶדֶת בּוֹ — מוּתָּר. וְהֵיכִי דָמֵי יָד סוֹלֶדֶת בּוֹ? אָמַר רַחֲבָא: כׇּל שֶׁכְּרֵיסוֹ שֶׁל תִּינוֹק נִכְוֵית.

Rav Naḥman bar Yitzḥak said: The opinion of the first tanna is to prohibit doing so. He explains the dispute in the following manner: According to the first tanna, with regard to oil, even if the heat is not so great that the hand spontaneously recoils from it, it is prohibited to heat it. He holds that oil is subject to the prohibition of cooking, and warming it is tantamount to cooking it. And Rabbi Yehuda came to say, leniently, that warming it is not tantamount to cooking it. And Rabban Shimon ben Gamliel came to disagree with Rabbi Yehuda and to say that oil is subject to the prohibition of cooking, and warming it is tantamount to cooking it. The Gemara questions: According to this explanation, the opinion of Rabban Shimon ben Gamliel is identical to the opinion of the first tanna. What is the difference between them? The Gemara answers: There is a practical difference between them in a case where this is done in a backhanded manner, i.e., not as it is typically done. According to the first tanna, it is totally prohibited to heat the oil, whereas according to Rabban Shimon ben Gamliel, it is permitted to heat the oil in a backhanded manner. Rav Yehuda said that Shmuel said that the halakha is: With regard to both oil and water, heating either one to the point where the hand spontaneously recoils from it is prohibited. Heating either one to the point where the hand does not spontaneously recoil from it is permitted. The Gemara asks: And what are the circumstances in which a hand spontaneously recoils from it? Not all hands are equal in their sensitivity to heat. The Sage, Raḥava, said: Any water that could cause a baby’s stomach to be scalded is considered water from which the hand spontaneously recoils.

(א) (א) המבשל וכו' - אחד המבשל את המאכל או את הסממנין או המחמם את המים [ר"ל אף דמים ראוי לשתיה בלא בישול וה"ה חלב מ"מ חייב כיון דמשתבח ע"י הבישול] ואחד האופה את הפת הכל ענין אחד הוא [דאפיה הוא מענין בישול אלא שאפיה הוא בפת ובישול הוא בשארי דברים]. המתיך אחד ממיני מתכות כל שהוא או המחמם את המתכות עד שתעשה גחלת ה"ז תולדת מבשל [שהרי בחימומו הוא מרככו שיהא ראוי לתקנו ולעשות ממנו כלי והרי הוא מרפה דבר הקשה] וכן הממסס את הדונג [שעוה] או את החלב או את הזפת והכופר והגפרית וכיוצא בהן ה"ז תולדת מבשל וחייב וכן המבשל כלי אדמה עד שיעשו חרס חייב משום מבשל [דקודם שמתחזק הכלי ונצרף בהכבשן ונעשה חרס כראוי מתבשל בה טיט של הכלי] וכן הנותן חתיכת עץ בתנור כדי שיתייבש וידוע שמתחלה הוא מתרפה ויוצא ממנו הלחלוחית חייב משום מבשל כללו של דבר בין שרפה גוף קשה באש או שהקשה גוף רך ה"ז חייב משום מבשל [רמב"ם בפ"ט] וע"כ יש ליזהר מאד שלא להניח עצים לחים על התנור ליבשן אחר שקיעת החמה דהוא חשש דאורייתא. הניח בשר ע"ג גחלים אם נצלה צליה גמורה כשיעור גרוגרת חייב ואפילו הגרוגרת אינה במקום אחד אלא מתלקט משנים ושלשה מקומות ג"כ חייב ואפילו אם לא היה צליה גמורה רק כמאב"ד ג"כ חייב אלא דבזה שאינו צליה גמורה צריך שיהיה הבשר צלוי משני הצדדים של הבשר דוקא וא"ל פטור [כן כתב הרמב"ם שם בהלכה ה'] וכ"ז הוא לענין חטאת אבל איסורא יש בכל גווני וכן מה שכתב הרמב"ם מתחלה לענין צליה השיעור גרוגרת הוא ג"כ רק לענין חיוב חטאת אבל איסור דאורייתא יש בכל גווני כדקי"ל חצי שיעור אסור מן התורה [שבת ע"ד גמ' ורש"י שם]:

If one cooks - Whether one cooks food or dyes or water [meaning even though water is fit to drink without cooking, and all the more so milk, still it is forbidden because it improves through cooking]...

The Mishneh Berurah learns that the case of water is mentioned by the Rambam to add the aspect to the Av of Bishul that improving the item is enough of a change to be the melacha.

(ב) מג"א ס"ק יג דתפוחים חיין לצולן אסור לא ידעתי מה כ"ש איכא הא הכא גרע לענין קירוב בשול הוי בשול גמור דאורייתא (ועיין מ"ש בגליון רסי' שס"ח) אלא דהכא הוי רק דרבנן כיון דנאכלים חיים וא"צ בשול כמ"ש הרמב"ם פ"ט הל' ג' אבל עכ"פ הבשול הוי בשול מעליא דבעלמא כה"ג חייב ואלו להניחה סמוך לתנור קודם שהוסק בעלמא הוי ג"כ דרבנן דבעידנא דנותן שם הפירות עדיין ליכא אור וכדקיי"ל א' נותן מים וא' נותן אור דהנותן מים פטור א"כ קיל יותר מהך דהכא ומה כ"ש הוא:

(2) Magen Avraham 13 - "raw apples are forbidden to be roasted" - I don't know what "all the more so..." can be said here? Here it is worse in terms of bishul it is forbidden on the Torah level to cook (see what I wrote on the margin in the beginning of Siman 368) rather over here it is only Rabbinically forbidden since they are eaten raw and they don't need cooking as the Rambam writes (Shabbos 9,3) though either way the cook is a proper cooking for in general if one did this [to something which needed to be cooked] it would be forbidden to do so on the Torah level....

Rabbi Akiva Eiger learns that something which is edible raw is not a Torah prohibition to cook it, even if it will improve through the cooking.
Minchas Cohen understands the Rambam that there is no issur even mdrabanan on something edible raw, unless you cook it on the fire, in which case it is forbidden mdrabanan because it is similar to cooking in that aspect [RAE said similar.] So where does the Rambam's issur dOraysa for cooking water come from?
The Eglei Tal learns that when it comes to water, hot water is a different item than cold water. His proof is that hot water is drunk for pleasure while cold water is only drunk to quench one's thirst. Therefore, when you heat up water, you are changing it significantly, and that is why you are forbidden to do so according to the Torah. This state of "hot water" is not around until you heat it up, so "hot water" is not "edible raw" and therefore heating it up, is the same significance of change as any food which needs to be cooked to be eaten.
According to this: unless there is a significant change you are not cooking mDoraysa. Although if you are putting the item on the fire then it will be forbidden mdrabanan because it is similar to the Av of cooking in that way. This argues with the Mishnah Berurah who holds that we see from water that even "slight improvement" is enough to transgress the Torah level prohibition.
[I wonder if this has to do with the climate? In hot weather drinking cold water can be refreshingly cool, whereas hot water would only be drunk to quench one's thirst. Obviously in cold climates, hot water can be enjoyed for the warmth it provides. Then there are the tea drinkers. How is the halacha determined? Based on whom?]
Different way to answer the Rambam according to Rabbi Akiva Eiger:

האופה - לא הוה במשכן דלא שייך אלא בפת ופת לא שייכא במלאכת המשכן אבל כולהו קמייתא הואי בסממנין של צבע תכלת וארגמן ותולעת שני ובגמרא פריך דשביק תנא מבשל דהוי בסממנים ונקט אופה ושיעורן של אלו כגרוגרת חוץ מחורש דבכל שהוא לקמן בפרק הבונה (שבת דף קג.):

...the amount of these [Melachos] [meaning the amount of item that you need to affect to be punished for transgressing the issur] is the size of a fig, with the exception of plowing, which is punishable even for a little bit as explained later on (on daf 103a).

Where does Rashi get this size from?

מַתְנִי׳ הַמּוֹצִיא תֶּבֶן — כִּמְלֹא פִי פָרָה. עָצָה — כִּמְלֹא פִי גָמָל. עָמִיר — כִּמְלֹא פִי טָלֶה. עֲשָׂבִים — כִּמְלֹא פִי גְדִי. עֲלֵי שׁוּם וַעֲלֵי בְצָלִים, לַחִיםכִּגְרוֹגֶרֶת, יְבֵשִׁים — כִּמְלֹא פִי גְדִי. וְאֵין מִצְטָרְפִין זֶה עִם זֶה מִפְּנֵי שֶׁלֹּא שָׁווּ בְּשִׁיעוּרֵיהֶן.

MISHNA: The mishna lists the measures in which various substances are significant and generally stored. One who carries out straw in a measure equivalent to a cow’s mouthful is liable. The measure that determines liability for etza is equivalent to a camel’s mouthful. Because it is a coarser food, he must carry out a greater amount in order to be liable. The measure that determines liability for ears of grain is equivalent to a lamb’s mouthful. The measure that determines liability for grass is equivalent to a goat’s mouthful, which is smaller than that of a lamb. The measure that determines liability for garlic leaves and onion leaves, if they are moist and fit for human consumption, is equivalent to a dried fig-bulk. A dried fig-bulk is the standard measure for human food. If the garlic leaves and onion leaves are dry, the measure for liability is equivalent to a goat’s mouthful. And none of these substances join together with one another to constitute a measure for liability because they are not equal in their measures.

לחין - הראויין לאדם כגרוגרת דזה שיעור לכל מאכל אדם לשבת אבל כמלא פי גדי לא דלחים לא חזו לגדי:

Moist ones - which are fit for man; the size of a fig, for this is the measure for all human food issues of Shabbos...

מַתְנִי׳ הַמּוֹצִיא אוֹכָלִים כִּגְרוֹגֶרֶת — חַיָּיב, וּמִצְטָרְפִין זֶה עִם זֶה מִפְּנֵי שֶׁשָּׁווּ בְּשִׁיעוּרֵיהֶן. חוּץ מִקְּלִיפָּתָן, וְגַרְעִינֵיהֶן, וְעוּקְצֵיהֶן, וְסוּבָּן, וּמוּרְסָנָן. רַבִּי יְהוּדָה אוֹמֵר: חוּץ מִקְּלִיפֵּי עֲדָשִׁין שֶׁמִּתְבַּשְּׁלוֹת עִמָּהֶן.
MISHNA: One who carries out a measure of foods fit for human consumption equivalent to a dried fig-bulk into a domain where carrying is prohibited on Shabbat is liable. And all those foods join together with one another to constitute that amount because they are equal in their measures. This amount is calculated without their shells, and their seeds, and their stems, and their bran, the husk that comes off of the wheat kernel when pounded, and their coarse bran that remains in the flour. Rabbi Yehuda says: None of the shells are calculated, except for the shells of lentils, which join together with the lentils to comprise the measure for liability because they are cooked and eaten with them.
מַתְנִי׳ הַמּוֹצִיא יַיִן — כְּדֵי מְזִיגַת הַכּוֹס. חָלָב — כְּדֵי גְמִיעָה. דְּבַשׁ — כְּדֵי לִיתֵּן עַל הַכָּתִית. שֶׁמֶן — כְּדֵי לָסוּךְ אֵבֶר קָטָן. מַיִם — כְּדֵי לָשׁוּף בָּהֶם אֶת הַקִּילוֹר. וּשְׁאָר כׇּל הַמַּשְׁקִין בִּרְבִיעִית, וְכׇל הַשּׁוֹפְכִין בִּרְבִיעִית. רַבִּי שִׁמְעוֹן אוֹמֵר: כּוּלָּן בִּרְבִיעִית, וְלֹא נֶאֶמְרוּ כׇּל הַשִּׁיעוּרִין הַלָּלוּ אֶלָּא לְמַצְנִיעֵיהֶן.
MISHNA: One who carries out undiluted wine from a private domain to a public domain or vice versa is liable only for a measure equivalent to the wine typically diluted in a cup. Pure wine was diluted with water. The measure that determines liability for carrying out wine is a measure suitable to be diluted for a significant cup of wine. The measure that determines liability for carrying out milk is equivalent to that which is swallowed in one gulp. The measure that determines liability for carrying out honey is equivalent to that which is used to place on a sore caused by chafing. The measure that determines liability for carrying out oil is equivalent to that which is used to spread on a small limb. The measure that determines liability for carrying out water is equivalent to that which is used to rub and spread on an eye bandage. And the measure that determines liability for carrying out all other liquids is a quarter of a log. And the measure that determines liability for carrying out all waste water is a quarter of a log. Rabbi Shimon says: The measure that determines liability for all liquids is a quarter of a log. He further stated: And all these measures were only stated with regard to those who store them. One indicates that he considers these liquids significant by storing them. One is only liable for carrying out an object that is significant to him. Others, for whom these measures are insignificant, are not liable for carrying them out.
We see here that the measurement for food is a fig size, so presumably for cooking the amount for which one would be punished is a fig amount.

(א) הָאוֹפֶה כִּגְרוֹגֶרֶת חַיָּב. אֶחָד הָאוֹפֶה אֶת הַפַּת אוֹ הַמְבַשֵּׁל אֶת הַמַּאֲכָל אוֹ אֶת הַסַּמְמָנִין אוֹ הַמְחַמֵּם אֶת הַמַּיִם הַכּל עִנְיָן אֶחָד הוּא. שִׁעוּר הַמְחַמֵּם אֶת הַמַּיִם כְּדֵי לִרְחֹץ בָּהֶן אֵיבָר קָטָן. וְשִׁעוּר מְבַשֵּׁל סַמְמָנִין כְּדֵי שֶׁיִּהְיוּ רְאוּיִין לְדָבָר שֶׁמְּבַשְּׁלִין אוֹתָן לוֹ:

(1) One who bakes [something] the size of a dried fig is liable. It is the same whether he bakes bread, cooks a food or herbs or heats up water – it all has one function. The [requisite] amount of heating water is enough to bathe a small limb. And the [requisite] amount of cooking herbs is enough that they be fit for the purpose for which they are being cooked.

(א) שיעור המחמם וכו'. כתב הרמ''ך לא ידעתי מאין הוציא וצ''ע, עכ''ל:

(1) The amount for one who heats something up etc - the Ramach writes "I don't know where he got this from and it needs further research".

The Rambam gives 2 amounts for heating up water: a dried fig's worth and enough to wash a small limb. So which one is it? Since washing a limb implies that the result of the cooking is that the water becomes usable for washing, the difference between the 2 measurements is: what is the purpose for which you are heating up the water? If it is for drinking the amount is a reviis. If it is for washing, it is the smallest amount of significant washing that you could do.
If so we can answer Rabbi Akiva Eiger's issue by saying that heating up water is for washing not eating that the Rambam holds you transgress the Torah prohibition.
All this was according to the Rambam, not all Rishonim agree with this though:

דזיתים מסקי הבלא - מכאן יש לאסור להניח גחלים תחת הקדרה אפי' יתן עליהן אפר אין להטמין קדרה עליהם שהרי הגחלים מעלין הבל למעלה כמו גפת של זיתים ויש תימה היאך אנו מטמינין על כירות שלנו שקורין אשטר"א ואע"פ שגורפין אותו הוא מוסיף הבל כמו גפת של זיתים ואומר ר"י שיש ליתן טעם לקיים המנהג דגבי גפת איכא למיחש שמא יטמין כולה בתוכה אבל בכירות שלנו לא שייך למיחש הכי ועוד יש שעושין חפירה גדולה ועושין בה בנין לבנים סביב מכל צד ומלמטה ומחמין אותה היטב וגורפין אותה ומטמינין בה את הקדרה ולא דמיא למטמין בדבר המוסיף הבל דלא אסרו אלא כשמטמין ומדביק סביב הקדרה דומיא דרמץ אבל תנור או חפירה שיש אויר בין הדפנות לקדרה אין לאסור יותר מהשהאה אף על פי שכל הקדרה בתוך התנור ורבינו ברוך פי' שיש לחלק בין תוספת הבל דגפת לתוספת הבל דכירה שלנו לפי שהכירה אין חומה אלא מחמת האש ולעולם מתקרר והולך אבל הגפת מוסיף הבל מעצמו אומר רבינו שמואל דמותר לשום תפוחים אצל האש סמוך לחשכה אע"פ שלא יוכלו לצלות מבע"י דנאכלין טפי כמו שהן חיין מתבשיל שנתבשל כמאכל בן דרוסאי דשרי ואע"ג דבפ"ק (ד' יג:) תנן אין צולין בשר בצל וביצה אלא כדי שיצולו מבע"י אע"פ שפעמים אוכלים בצל חי מ"מ אינו ראוי לאכול חי כמו תפוחים ורוצה להפיג חריפותן ולמתקן בבישולו ואם הטמין תפוחים עם הקדרה אסור להחזיר כרים וכסתות על הקדרה שעמהן או להוסיף על אותן כרים אע"ג דאם אין שם תפוחים מותר להחזיר כדתנן בפירקין (ד' נא.) כסהו ונתגלה מותר לכסותו ולהוסיף נמי אמרינן בגמרא (שם.) אם בא להוסיף מוסיף כשיש עם הקדרה תפוחים אסור דאם מחזיר קודם שנתבשלו נמצא מבשל בשבת ומה שנהגו להסיר הקדרה מעל הכירה ומניחין על הקרקע עד שיגרפו הכירה שמא סבירא לן כהך לישנא דריש כירה (לעיל דך לח:) דשרי חזקיה משמיה דאביי הניחו על גבי קרקע אם דעתו להחזיר:

... Rabeinu Shmuel writes that it is permitted to place apples near the fire close to nightfall [at the start of Shabbos] even though it will not be able to roast while it is still day [ie before Shabbos begins] for they are more commonly eaten raw than a food which is cooked mb'd which is permitted. Even though in the first perek (13b) the Mishnah states that one may not roast meat, an onion, or an egg unless they will be roasted while it is still daytime, even though sometimes onions are eaten raw, still it is not as fit to eat as a [raw] apple so he wants to blunt the sharpness and to sweeten [the onion] by cooking it. If one insulate apples with a pot it is forbidden to place pillows or blankets on the pot that they [the apples] are with, or to add to the pillows, even though if there weren't any apples it would be permitted to put back the pillows and blankets as it says in the mishnah (51a) if he covered it and it became uncovered one is permitted to cover it. To add to the covering we also say in the gemara "if one wants to add [pillows] he may add. [Though] when there are apples with the pot it is forbidden for if he would return the insulation before they cooked he would be cooking on Shabbos [since the insulation that he added would cause the cooking of the apples to be completed.]......

We see that Tosafos holds that there is a prohibition to cook the apples from raw to "roasted". [This would seem to fit with the Mishnah Berurah of earlier, who held that improving a food is called cooking too.]
The Ohr Zaruah quotes that the Riva held there is cooking on wine. [Another Rishon who holds that the change does not need to be major, even slight improvement is enough to transgress cooking.]
The Kol Bo also holds that there is a Torah prohibition of cooking on a food that is edible raw. [A 3rd Rishon who holds this way.]