Shabbos 39a - Defining Bishul

The gemara doesn't directly discuss the definition of bishul on a Torah level. Tosafos d.h. kol, says that anything that has been fully cooked prior to shabbos, would be permitted to put in a kli rishon on shabbos. The Biur Halacha (318:4) points out that this implies that anything which is not fully cooked would be prohibited on a Torah level to advance its cooking beacuse it would be considered a violation of bishul. This is the opinion of the Rambam and Shulchan Aruch that advancing cooking, even after it is already cooked to ma'achal ben drusai would be a violation of cooking. Therefore, even if the food is still hot, one cannot do anything to advance its cooking i.e. stirring or covering the pot. Furthermore, the Shulchan Aruch implies that even if something is fully cooked and cools below yad soledes bo, raising the temperature to above yad soledes bo would also constitute a violation of cooking.

However, the Biur Halacha points out that all the Rishonim who permit chazara on something which is ma'achal ben drusai, must hold that after the point of ma'achal ben drusai there is no longer a violation of cooking. The Ran (17b) writes this explicitly - דכיון שנתבשל כמאכל בן דרוסאי לית ביה משום מבשל דאורייתא. Tosafos 36b d.h. chamin, also permits chazara on something which is ma'achal ben drusai, indicating that there is no cooking violation at that point. The Biur Halacha points out that there are many rishonim on both sides, and therefore it is worth being machmir for an issur d'oraysa.

It seems that both opinions hold that bishul is not defined at a finite point. Meaning, unlike water where the violation of bishul occurs when something is brought from the temperature of below yad soledes to above yad soledes, cooked foods have no finite point. Rather, until something is "cooked" which according to some is fully cooked (shulchan aruch) and according to others is ma'achal ben drusai (ran), there is a Torah violation of cooking to do anything that will advance or hasten the process, even if it never reaches the point of fully cooked or ma'achal ben drusai.

It seems from shulchan aruch that liquid foods that require cooking are bound to both definitions of cooking. If they are below yad soledes and one raises their temperature to above yad soledes, that constitutes cooking. And if they are already yad soleded but not yet fully cooked and one raises the temperature by covering the pot to hasten the cooking, that would also constitute cooking.