Ben Drusaiy-Half cooked foods:
The Sages forbade leaving specifically less than half cooked food [i.e., Ben Drusaiy] on an uncovered flame from before Shabbos due to worry that by less than half cooked food, one may come to raise the flame on Shabbos in order to hasten the cooking. If, however, the food is more than half cooked before Shabbos starts, which is before Shekia and not candle lighting, then the Sages did not suspect that one would instinctively raise the flame. Accordingly, one may leave food that is more than half cooked over an open flame over Shabbos, and the Shehiyah restrictions do not apply in such a case. Furthermore, Bedieved [or in a time of need] one may be lenient if the food was 1/3rd cooked before Shabbos, and the prohibition only applies if the food is not even 1/3rd cooked. [Nonetheless, many are stringent not to leave food over an uncovered fire into Shabbos, even if the food is half cooked or fully cooked, and only do so if further cooking will damage the food, i.e. Mitztameik Vera Lo.]
If one accepts Shabbos early, must the food be half cooked before he accepts Shabbos, or does it suffice for it to be half cooked before sunset, the time that Shabbos actually begins?
As long as the food will be half cooked before sunset it may be placed on an uncovered fire before one accepts Shabbos.
Does all the food in the pot need to be half cooked, or does it suffice if most of the food in the pot is half cooked?
All the food in the pot needs to be half cooked, as otherwise the suspicion that one may come to raise the flame to hasten the cooking is still applicable to the minority part that is not yet cooked.
How does one measure if a food is half cooked?
Some Poskim say that it is determined in terms of the quality of the food, meaning that only if the food has reached half the point of its desired quality is it considered half cooked, even if it has already cooked for half the time that it needs. Other Poskim however say that it is determined by if the pot has stayed on the fire for half its required time. The Kaf Hachaim writes that it is determined as follows: As long as one is already able to eat the food at a pressing situation, such as when he is in a hurry, then it is considered half cooked.
 1st opinion and final custom in Admur 253:8-9; Second opinion in Michaber 253; Rashi; Riy
Other opinions-Forbidden unless is Mitztameil Vera Lo: There are opinions who say that [the Shehiyah restrictions apply] even [by] food which has been cooked to the point of the food of Ben Drusaiy. [Furthermore, it applies] even if it has been fully cooked but [further cooking will still make] it condenses [in a way that] will benefit the food [i.e. Mitztameik Veyafeh Lo]. Meaning [not that it will benefit the actual taste of the food per say but] that it’s [further cooking] is beneficial to the person [cooking it] and he is pleased with the [further] condensing of the food. The reason for why if the food can still condense it is prohibited according to this opinion is because perhaps one will stoke the fire on Shabbos to make it ready more rapidly, or in order so it will be condensed. For this reason, it has the same laws, for all matters, just as if it had begun to cook [before Shabbos] but was not yet cooked to the point of the food of Ben Drusaiy, of which was explained above [that it has restrictions in regard to which ovens it may be placed on before Shabbos]. [2nd opinion in Admur 253:9 and 254:14; Rif; Rambam; first opinion in Michaber] Practically, the custom has already spread to be lenient like the first opinion [that if the food has been cooked to the point of Ben Drusaiy, then it may be left on any flame over Shabbos]. [Admur 253:9; Rama 253:1] Some Poskim, however, rule that one is to initially be stringent like the stringent opinion brought here. [Biur Halacha 253 “Venahagu Lihakel”]
Definition of Mitztameik Veyafeh Lo: Rav Farkash explains in Shabbos Kehalacha [Vol. 1 page 293] that the definition of further cooking being beneficial is solely dependent on the person cooking it. Meaning that if the person cooking wants the food to condense, such as that he desires it to be tastier, then it is considered beneficial, while if the person does not want it to condense, such as that he wants the portions to look nice and plump, rather than condensed and shriveled, even though doing so gives it a better taste, then nevertheless for him such further cooking is not considered beneficial, and would be allowed according to all.
 See next footnote for opinions who rule that the prohibition is only by food that is not 1/3rd cooked.
 Ben Drusaiy was the name of a bandit in the Talmudic era who would eat his food prior to it being fully cooked. [Admur 253:1 in parentheses] He was always on the run to make the next robbery of his victims that passed by his wooded area, so he never had time to properly cook his food. He had a lot of customers. He would thus eat his food when it was only half cooked. Whenever the term “Ben Drusiay” is used throughout Halacha in general and this lesson in particular, such as “If the food was Ben Drusiay” it refers to half cooked.
How much is Machel Ben Drusaiy? The measurement of [cooking for] the food of Ben Drusaiy, some opinions [i.e. Rambam] say is about a third cooked from the amount of cooking which is common for everyone. Others [i.e. Rashi, Riy, Rashba, Maharam and other Poskim, brought in Kuntrus Achron 2], however, say that it is half cooked [from the amount that people consider fully cooked]. Practically the law is that even though by Rabbinical prohibitions we rule leniently [when there is a dispute], nevertheless due to the severity of Shabbos one needs to be careful initially that from before Shabbos the food is cooked half the amount of its required cooking, [in a case that] none of the other explained leniencies apply. However, if after the fact [one had placed the pot on the fire without it having been ½ cooked] then the food is permitted in all cases, if it had cooked a third of its required cooking from before Shabbos. [Admur 253:13]
 The reason: The reason for this is because once it has become half cooked, we are no longer worried he will stoke the coals, as even without stoking, the food is able to be fully cooked for the night meal being that it has already cooked halfway from before Shabbos. As well [another reason is] since the food has been [half] cooked from before Shabbos, the food is already [cooked enough to be] fit to be eaten in a pressing situation, [and we thus do not suspect that one will come to stir the coals]. [Admur ibid]
 Elya Rabah; Mishneh Berurah 253:38; Shemiras Shabbos Kehilchasa 1:63; Shabbos Kehalacha 1 p. 315 that in a pressing situation one may be lenient to leave the food on a revealed fire if it is only 1/3 cooked.
 2nd opinion and final ruling of Admur ibid that Bedieved we are lenient if the food is 1/3 cooked. See previous footnotes
 2nd opinion in Admur ibid, brought in previous footnote; Biur Halacha 253 “Venahagu Lihakel”; See Shabbos Kehalacha Vol. 1 p. 320 and 339 that although as stated above the custom is not like this opinion, nevertheless today many are stringent to make sure that the food is fully cooked on Erev Shabbos, and in addition to this they still make sure that the flame is covered. This is done in order to satisfy all existing opinions on the matter. In the supplements [page 339] he as well seeks basis for this in the wording of the Alter Rebbe. Nevertheless, according to the Alter Rebbe there is no obligation for one to follow this stringency and at best it is only that there is room for stringency even in accordance with the Alter Rebbe’s opinion.
 Shabbos Kehalacha Vol. 1 pages 313-314
 This is inferred from the wording of the Alter Rebbe in Halacha 1 “if by the time the night [of Shabbos] begins the dish is still not cooked to the point of Ben Drusaiy. This is based on the ruling of the Magen Avraham 253:26. Regarding the question why the decree applies only by sunset and not beforehand by Tosefes Shabbos, Rav Akivah Eiger answers that only a single moment before Shabbos is Biblically required for Tosefes Shabbos. The Peri Megadim answers that the Sages only made their decree when the laws of Kareis and Sekilah apply on Shabbos, and thus since Tosefes Shabbos is only a positive command, it does not apply then.
 Shabbos Kehalacha Vol. 1 page 314
 Shabbos Kehalacha Vol. 1 page 314-316
 Sheivet Haleivi, and Kloizenberger Rebbe, based on Shut Maharshag
 Chazon Ish, based on Peri Megadim
 Admur 253:28