Chicken and Cheese From what origin point are we approaching the question? We have a limited capacity to change tradition.
(יט) רֵאשִׁ֗ית בִּכּוּרֵי֙ אַדְמָ֣תְךָ֔ תָּבִ֕יא בֵּ֖ית יְהוָ֣ה אֱלֹהֶ֑יךָ לֹֽא־תְבַשֵּׁ֥ל גְּדִ֖י בַּחֲלֵ֥ב אִמּֽוֹ׃ (ס)

(19) The choice first fruits of your soil you shall bring to the house of the LORD your God. You shall not boil a kid in its mother’s milk.

"The Observant Life: The Wisdom of Conservative Judaism for Contemporary Jews" by Martin Cohen, Michael Katz and wanted to share this quote with you.

"Three times, at Exodus 23:19, Exodus 34:26, and Deuteronomy 14:21, Scripture ordains that one may not eat the flesh of a kid that has been cooked in its own mother’s milk. The oral tradition connected with these verses vastly expands the concept to include three major areas of prohibition: dairy and meat products may be neither eaten nor cooked together, nor may one derive even ancillary benefit from such a forbidden mixture (MT Hilkhot Ma·akhalot Asurot 9:1). Although there is no specific source in the Bible or the Talmud that clearly states the reason for this supremely stringent policy of separation, many ideas have been suggested throughout the ages. Maimonides, for example, writing in the Guide for the Perplexed (III 98), took a historical approach, writing that he imagined these laws to be a kind of reaction to the pagan custom of boiling a kid in its mother’s milk, which practice he imagined tied to pagan fertility rites. More recently, however, Professor Edward L. Greenstein of Bar-Ilan University has taken a more anthropological approach, writing that the prohibition" "which is meant to sustain life, may not be turned into a means of preparing an animal for eating. A clear distinction must be made between life, which is godly, and death. The post-biblical Jewish tradition underscores the distinction by broadening it: not only milk, but all dairy products and the utensils used for serving them must be kept apart from meat products and utensils” (Etz Hayim, p. 1461). Whatever its rationale, however, the rabbis understood the law to apply to situations far beyond the literal meaning of the biblical verses. The flesh of mammals and birds are both considered meat. Furthermore, even foods made with meat additives or meat by-products are considered meat in this context, even if there are no actual pieces of meat present."
(ג) כֹּ֣ל ׀ מַפְרֶ֣סֶת פַּרְסָ֗ה וְשֹׁסַ֤עַת שֶׁ֙סַע֙ פְּרָסֹ֔ת מַעֲלַ֥ת גֵּרָ֖ה בַּבְּהֵמָ֑ה אֹתָ֖הּ תֹּאכֵֽלוּ׃
(3) any animal that has true hoofs, with clefts through the hoofs, and that chews the cud—such you may eat.
(ט) אֶת־זֶה֙ תֹּֽאכְל֔וּ מִכֹּ֖ל אֲשֶׁ֣ר בַּמָּ֑יִם כֹּ֣ל אֲשֶׁר־לוֹ֩ סְנַפִּ֨יר וְקַשְׂקֶ֜שֶׂת בַּמַּ֗יִם בַּיַּמִּ֛ים וּבַנְּחָלִ֖ים אֹתָ֥ם תֹּאכֵֽלוּ׃ (י) וְכֹל֩ אֲשֶׁ֨ר אֵֽין־ל֜וֹ סְנַפִּ֣יר וְקַשְׂקֶ֗שֶׂת בַּיַּמִּים֙ וּבַנְּחָלִ֔ים מִכֹּל֙ שֶׁ֣רֶץ הַמַּ֔יִם וּמִכֹּ֛ל נֶ֥פֶשׁ הַחַיָּ֖ה אֲשֶׁ֣ר בַּמָּ֑יִם שֶׁ֥קֶץ הֵ֖ם לָכֶֽם׃
(9) These you may eat of all that live in water: anything in water, whether in the seas or in the streams, that has fins and scales—these you may eat. (10) But anything in the seas or in the streams that has no fins and scales, among all the swarming things of the water and among all the other living creatures that are in the water—they are an abomination for you
(יג) וְאֶת־אֵ֙לֶּה֙ תְּשַׁקְּצ֣וּ מִן־הָע֔וֹף לֹ֥א יֵאָכְל֖וּ שֶׁ֣קֶץ הֵ֑ם אֶת־הַנֶּ֙שֶׁר֙ וְאֶת־הַפֶּ֔רֶס וְאֵ֖ת הָעָזְנִיָּֽה׃
(13) The following you shall abominate among the birds—they shall not be eaten, they are an abomination: the eagle, the vulture, and the black vulture;

רַבִּי עֲקִיבָא אוֹמֵר, חַיָּה וָעוֹף אֵינָם מִן הַתּוֹרָה, שֶׁנֶּאֱמַר, לֹא תְבַשֵּׁל גְּדִי בַּחֲלֵב אִמּוֹ, שָׁלשׁ פְּעָמִים, פְּרָט לְחַיָּה וּלְעוֹף וְלִבְהֵמָה טְמֵאָה. רַבִּי יוֹסֵי הַגְּלִילִי אוֹמֵר, נֶאֱמַר (דברים יד), לֹא תֹאכְלוּ כָל נְבֵלָה, וְנֶאֱמַר (שם), לֹא תְבַשֵּׁל גְּדִי בַּחֲלֵב אִמּוֹ. אֶת שֶׁאָסוּר מִשּׁוּם נְבֵלָה, אָסוּר לְבַשֵּׁל בְּחָלָב. עוֹף, שֶׁאָסוּר מִשּׁוּם נְבֵלָה, יָכוֹל יְהֵא אָסוּר לְבַשֵּׁל בְּחָלָב, תַּלְמוּד לוֹמַר בַּחֲלֵב אִמּוֹ, יָצָא עוֹף, שֶׁאֵין לוֹ חֲלֵב אֵם:

Rabbi Akiva says, "Wild animals and birds are not [forbidden in mixtures with milk] from the Torah, as it says, 'You may not cook a kid in its mother's milk,' three times. It exempts the wild animal, the bird, and the prohibited domesticated animal." Rabbi Yose HaGelili says, "It is stated, 'You may not eat any nevelah [an improperly slaughtered animal of a permitted species],' (Deuteronomy 14:21) and it is stated, 'You may not cook a kid in its mother's milk,' (Ibid.). That which [may become] forbidden as nevelah is forbidden to cook with milk. A bird, which [may become] forbidden as a nevelah, one might think that it should be forbidden to cook with milk, [but] the Torah says, "In its mother's milk." This excludes a bird, which does not have mother's milk.

אלא חו"ל משום דליכא למיגזר הוא אבל הכא אי שרית ליה לאסוקי עוף וגבינה אתי לאסוקי בשר וגבינה ומיכל בשר בחלב דאורייתא מתקיף לה רב ששת סוף סוף צונן בצונן הוא אמר אביי גזירה שמא יעלה באילפס רותח
Rather, [Challah] of outside the land of Israel [is not prohibited] because there is nothing to decree upon. However, here, if you allow fowl and cheese together, you might allow beef and cheese, and you would, therefore, be eating a Torah prohibited mixture of flesh and cheese. Rav Sheshet challenged him: Either way, [the combination of cheese and chicken] is cold with cold. Abayei said: It's a preventative measure, lest you bring it up in a boiling tray.
מתני׳ העוף עולה עם הגבינה על השולחן ואינו נאכל דברי ב"ש וב"ה אומרים לא עולה ולא נאכל א"ר יוסי זו מקולי ב"ש ומחומרי ב"ה
Fowl may be placed on a table with cheese but may not be eaten [together], these are the words of Bait Shammai. However, Bait Hillel say It may not be placed on [the table] nor eaten [together]. Rabbe Yose said: This id of the leniencies of Bait Shammai and the stringencies of Bait Hillel.

(ג) אֵינוֹ נוֹהֵג אֶלָּא בִּבְשַׂר בְּהֵמָה טְהוֹרָה בַּחֲלֵב בְּהֵמָה טְהוֹרָה, אֲבָל בְּשַׂר טְהוֹרָה בַּחֲלֵב טְמֵאָה, אוֹ בְּשַׂר טְמֵאָה בַּחֲלֵב טְהוֹרָה, מֻתָּרִים בְּבִשּׁוּל וּבַהֲנָאָה. וּבְשַׂר חַיָּה וָעוֹף, אֲפִלּוּ בַּחֲלֵב טְהוֹרָה, מֻתָּר בְּבִשּׁוּל, וּבַהֲנָאָה; וְאַף בַּאֲכִילָה אֵינוֹ אָסוּר, אֶלָּא מִדְּרַבָּנָן. אֲבָל דָּגִים וַחֲגָבִים, אֵין בָּהֶם אִסוּר, אֲפִלּוּ מִדְּרַבָּנָן. הַגָּה: וְנָהֲגוּ לַעֲשׂוֹת חָלָב מִשְּׁקֵדִים וּמַנִּיחִים בָּהּ בְּשַׂר עוֹף, הוֹאִיל וְאֵינוֹ רַק מִדְּרַבָּנָן. אֲבָל בְּשַׂר בְּהֵמָה, יֵשׁ לְהַנִּיחַ אֵצֶל הַחָלָב שְׁקֵדִים, מִשּׁוּם מַרְאִית הָעַיִן, כְּמוֹ שֶׁנִּתְבָּאֵר לְעֵיל סִימָן ס''ו לְעִנְיַן דָּם (ד''ע).

(3) The law only applies with meat from a pure [kosher] animal and milk from a pure animal, but meat of a pure animal in milk that is impure, or meat from an impure animal in pure milk, has the eating prohibition but not the restriction on cooking or deriving benefitting from. Meat of a wild animal and of fowl, even in milk which is pure, is permitted to cook and benefit from; and even eating is only prohibited rabbinically. Fish and grasshoppers are not prohibited [with milk] even rabbinically. We make milk from almonds and place bird meat in it, since [milk and bird meat] is only rabbinically [forbidden]. But with meat from a domesticated animal, place almonds next to the milk, so that people don't misunderstand. This is as we said above, in chapter 66.

Dr. David Kraemer, Professor of Talmud, JTS:
“In the ancient world, meat was eaten with relative rarity, primarily for special occasions,” Kraemer told me. “For more common special occasions, such as the Sabbath, ‘smaller’ meat would have been most common, and that was typically fowl. So people simply thought of and spoke of fowl as meat. Since this is the way people thought of it, this is the way the rabbis categorized it.”
מתני׳ כל הבשר אסור לבשל בחלב חוץ מבשר דגים וחגבים ואסור להעלות עם הגבינה על השלחן חוץ מבשר דגים וחגבים
Mishnah : One is forbidden to cook any kind of flesh in milk except fish and locusts. And one is forbidden to bring [the flesh] up onto the table with cheese, except fish and locusts.
What is the difference between a fish and a chicken? Animals and birds are subject to the laws of Kosher slaughter, fish are not. A practical difference explained in the Talmud.
אמר שמואל השולה דג מן הים כיון שיבש בו כסלע חייב א"ר יוסי בר אבין ובין סנפיריו אמר רב אשי לא תימא יבש ממש אלא אפילו דעבד רירי
Shmuel said: With regard to one who removes a fish from the sea, when an area on the skin of the fish has dried up the size of a sela, he is liable. A fish in that condition cannot survive, and therefore the individual who removed it from the water is liable for killing it. Rabbi Yosei bar Avin said: That is so as long as the skin that dried is between its fins. Rav Ashi said: Do not say that this halakha applies only in a case where it actually dried. Rather, it applies even if the fish has dried to the extent that mucus has formed, and if one were to touch that area it would stick to his fingers.
On this question let's just say the chicken has flown the coop