daf bit: Menachot 57
The g'mara on today's daf discusses the prohibition of cooking on Shabbat.
If a man placed meat on the coals on Shabbat and it was roasted, he is
liable. How much has to be cooked? After discussion about whether it
has to be cooked through or only on one side to count, Raba said that if
it was roasted in one spot the size of a dried fig that is sufficient.
There is further discussion in which either Raba or Rabina said that
if it is roasted in two or three spots that together add up to the size
of a dried fig, that too is a violation -- it doesn't have to be a single
area. Rabina then compares this to boring a hole on Shabbat, which is
prohibited regardless of the size of the hole, but the difference in
these cases is not addressed here. (57a)
I had assumed that any amount of cooking would be prohibited. Of course, this might not be the last word on the subject.

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Talmudically, I'm not sure. Halacha today is that you can't; you can keep food warm that is already cooked, but you can't drop it onto the fire a minute before Shabbat and eat later.
This is what almost all observant Jews I know do -- cook the meal Thursday night or Friday afternoon, then set up some warming device on a timer. For example, Shabbat morning I take the crock-pot liner or pan out of the fridge and put it in/on a heating unit, which a timer will turn on after I've left. Or I could leave something in the oven on very low heat. There is a device ("blech") for keeping things warm on the stove-top too, but I'm not comfortable leaving a flame running through Shabbat (gas stove, not electric).
We've got work-arounds, in other words. :-) But the core prohibition of cooking on Shabbat is maintained. Keeping things warm isn't cooking.
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Most of the Shabbat prohibitions have a minimum amount d'oraita. In theory, if you lined up enough yidden and each wrote one letter, they could write out all of War and Peace on shabbat and only be in violation of a rabbinic fence.
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