The Shulhan Aruch, amidst his discussion of the laws of Shabbat, codifies the prohibition of “Hagasa,” stirring food that has not been fully cooked. If a pot over the fire contains food that has not been completely cooked, one may not stir the food, as this accelerates the cooking process. Stirring food in such a situation transgresses a Torah violation. Moreover, even if the pot has been removed from the fire, one may not stir the food (assuming it has not been fully cooked) so long as it is hot, at or above the temperature of “Yad Soledet Bo” (the point where one’s hand would immediately recoil on contact).
If the food has been fully cooked, Halacha nevertheless forbids stirring it over a fire (based on the ruling of the Ritba); once the pot has been taken off the fire, one may stir the food, since it has been fully cooked.
However, the Halachic authorities debate the question of whether one may serve fully cooked food from a pot on a fire or on a blech (metal sheet covering the flame). Generally, we have fully cooked food on the blech when Shabbat begins, and this food is served at the Friday night meal. May one take food directly from the pot on the blech and transfer it to a serving dish, or must he first remove the pot from the blech and only then serve?
Some authorities, including the Ben Ish Hai (Rabbi Yosef Haim of Baghdad, 1833-1909), in his work Rav Pe’alim, and Hacham Ben Sion Abba Shaul (Israel, 1923-1998), in his Or Le’sion, rule stringently in this regard. In their view, dishing out food is no different from stirring with respect to this Halacha, and thus even fully cooked must be removed from the fire before food is taken from it. (These authorities allow serving fully cooked food under certain circumstances, but as a rule they are stringent on this issue.) Hacham Ovadia Yosef, however, in several places (Kol Sinai, and Yabia Omer, vol. 10), rules leniently, and distinguishes serving from stirring. Even though one may not stir fully cooked food over the fire, it is permissible to dish out fully cooked food over a fire. This also appears to have been the position of the Hazon Ish (Rabbi Avraham Yeshaya Karelitz, Lithuania-Israel, 1879-1954). Thus, even though one may not stir fully cooked food over a fire on Shabbat, it is permissible to dish out fully cooked food directly from the blech to put it into a plate or serving dish.
Summary: One may not stir on Shabbat food that is not fully cooked, neither on the fire (even with a blech) nor after it has been removed from the fire, so long as it is hot. If the food is fully cooked, one may stir it once it has been removed from fire or blech. While it is still on the blech, one may not stir it, but he may dish food out from the pot.