I wonder if eating a chicken food in which its ingredients take milk or cheese is allowed. (For example: Eat a Chicken Stroganoff; Or a chicken stew with sour cream ...)
Well there is no way to find in the Halacha an approval to eat chicken with dairy. Here is a fast and short answer I found in google randomly (in Hebrew and with a translation).
נאמר בתורה 3 פעמים "לא תבשל גדי בחלב אמו" ולמדו מכך חכמים: א. איסור לבשל חלב ובשר. ב. אסור לאכול חלב ובשר. ג. אסור להנות מכך. מדובר בבשר בהמה טהורה ולא בעוף. אך גזרו חכמים, שכיוון שאין בני אדם מבדילים בין בשר עוף לבשר בהמה ושניהם נקראים "בשר", על כן גזרו ואסרו אכילה של עוף עם חלב.
כל טוב אבינח ברנר
It is written in the bible 3 times "Do not cook a (baby) goat with the milk of his mother", and our "Chachamim" (The ancient Rabbi's who wrote the the talmud which our Halacha (law) is based on) learned from this three things:
Not to cook meat with milk.
Not to eat milk and meat.
Not to have any enjoyment from cooking meat and milk.
This is written about a "Holy" mammal and not chicken, But our "Chachamim" gave a boundary to this law (not to eat chicken) since people call meat of a mammal 'meat' and also the meat of chicken 'meat' and do not have a big recognition between the to types of meat.
Also by Rabbi Kook Judaism is trying to aim the Jews (and humanity) towards the direction to stop eating meat. From this perspective one reason why G-d commanded us not to eat meat and milk together is create in our mind set a recognition between killing an animal and taking its milk, you can not mix these two because morally they are very different! Therefore it is understood why also we can not eat chicken and milk, this to is a killing of an animal and a "theft" of the milk. (This you can find in his book of "Vision of Vegetarianism and Peace". Here is the Hebrew version).
Although it might seem so but it is not recommended by Rabbi Kook to become a full vegetarian he him self ate meat.