The cooking prohibition involved in reheating foods:

All foods that fulfill all the following conditions do not contain a cooking prohibition upon reheating them:

1. Fully cooked:[1] The food must be fully cooked, [to the point that it requires no further softening, or change of texture, through cooking]. If the food is not fully cooked, even if it is half cooked, it contains Biblical cooking prohibition against cooking it any further. It is hence forbidden to heat up such food on Shabbos.
2. Dry: [2] The food must be completely[3] dry and not contain any moisture at all. This means that a solid food must be dry to the point that it is not Tofeiach Al Menas Lehatfiach[4], which means that it’s moisture cannot wet one’s finger enough to make a further finger wet.[5] If the food contains cold moisture past this point, and certainly if the food contains actual cold liquid, it is forbidden to heat it on Shabbos due to the cooking prohibition, even if the food was fully cooked before Shabbos.
3. Does not melt:[6] The food may not contain congealed solids which will melt with the heat. If the food contains congealed solids that will melt with the heat, it is forbidden to be heated on Shabbos even if the food was previously cooked or roasted.

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[1] Admur 253:18; 318:10; Rama 253:2; Michaber 318:4; M”B 253 Biur Halacha “Vedavka Shehatvshil”; Biur Halcha 253:2 “Mutar Lehachziro”

Other Opinions: Some Rishonim rule that further cooking of a food that has reached Ben Drusaiy is permitted on Shabbos. [See Biur Halacha ibid and 318:4; Even Yisrael 8:28; Piskeiy Teshuvos 253:18]

[2] 2^nd opinion and final ruling and custom in Admur 318:9; Michaber 318:4; Rama 318:15; Rashi; Rabbeinu Yonah; Rosh

Other opinions: Some Poskim rule that even if [the precooked liquid] has [already] completely cooled down there is no [prohibition in] cooking it after [it is having been previously] cooked. [2^nd opinion in Admur ibid; Rama 318:15 ; Rashba; Ran; Magid Mishneh] The custom is to be stringent like the first opinion. [Admur ibid; Rama 318:15]

[3] Admur 253:19; 318:11; 318:17; Chasam Sofer 74; Divrei Nechemia p. 36 in Hosafos; Ketzos Hashulchan 124 footnote 37; Igros Moshe 4:74:7; Or Letziyon 2:30:13; Rav Yaakov Yosef severely negated the lenient opinion of Rabbeinu Yerucham and told Sefaradi askers they cannot rely on it Chas Veshalom even if there is a small amount of gravy on fish.

Background:

253:19: [Furthermore] even if the food [which one wants to return] is completely cooked but it has already completely cooled down, then it is forbidden to return it in all cases, even if it is a type of food which reheating it does not consist of a prohibition of cooking, such as is the case with a dry food that has no liquid at all that is fully cooked,

318:11: Dry food which contains no liquid at all is [permitted to be] cooked after having been previously cooked as long as it was already fully cooked, even if it completely cooled down.

318:17: Nevertheless, ox meat is forbidden to place inside a Keli Rishon [that is off the fire] as although the meat [itself] will not cook, nevertheless the moisture that is on it will cook. Although if [the meat] is completely dry and does not have on it any moisture at all [then it is allowed].

Chasam Sofer 74: “Only if the coffee is completely dry and contains no moisture at all”

Divrei Nechemia p. 36: “This opinion [if the Minchas Kohen] is Muktzah Min Hadaas and is based on a mistaken Nussach in the Rabbeinu Yerucham.

Other opinions: Some Poskim rule that so long as majority of the food is dry [solid] it is permitted to place it on the fire even if it containes a minority of liquid. [Nussach of Rabbeinu Yerucham in Beis Yosef 253; Minchas Kohen Mishmeres Hashabbos 2:2; Iglei Tal Ofeh 34; P”M 253 M”Z 13; Kaf Hachaim 318:62] This is because when majority of the food is a solid one does not have in mind to cook the liquid and in such a case one may even initially rely on those Poskim that are lenient to say there is no Bishul after Bishul by liquids. [Iglei Tal ibid]

[4] Ketzos Hashulchan 124 footnote 37

[5] 613:14

[6] Admur in Siddur; Levush 318:16; Taz 318:20; See Shabbos Kehalacha p. 148-151

Other opinions: Some Poskim rule there is no Bishul after Bishul by a precooked dissolvable item. [Admur 318:11; 318:25; 318:31; Michaber 318:16; M”A 318:40]