The classic prohibition of Bishul (cooking) on Shabbat consists of applying heat to a completely raw food. What is the Halacha regarding heating a food which was already partially cooked from before Shabbat? Such a food is known in the Gemara as a "Ma’achal Ben Drusai," referring to a food which is either half or one-third cooked, but edible to some people.
This question is a Mahloket Rishonim (dispute between the early authorities). Rambam (Shabbat 9:3) maintains that it is a Torah prohibition to advance the cooking of a partially cooked food. Maran adopts this strict position of the Rambam.
On the other hand, the Ramban and Rashba hold that it is permitted. They bring a proof from the Mishna which states that anything that was immersed in hot water before Shabbat may be immersed in hot water again on Shabbat. The intent of this Halacha is that there is no prohibition to cook foods that were already cooked. Yet, the fact that the Mishna used the example of items immersed in water, which implies not a full cooking process, as opposed to simply saying, "That which was cooked before Shabbat…" indicates that even foods that were partially cooked are exempt from the prohibition of Bishul.
Even though the Halacha is in accordance with Rambam and Maran, Hacham David writes a Chidush in Halacha Berura that one may instruct a non-Jew to continue cooking an item that was already partially cooked. Since doing so is subject to a Mahloket Rishonim, the Halacha allows instructing a non-Jew which is always only M’drabanan.
It is prohibited to cook or heat a food which was already partially cooked before Shabbat. However, one may instruct a non-Jew to do so.