Someone cooked on the Sabbath for the sake of a dangerously ill person and added extra for himself as well. According to the Ran, this is a Torah prohibition. Is this the prohibition of “adding more than the required amount,” or the prohibition of cooking on the Sabbath?
If it is the prohibition of cooking on the Sabbath, is he liable for it, or exempt from punishment but still forbidden by Torah law?

I don’t recall any source being cited for this prohibition itself (especially since there are opinions that it is rabbinic). So it is not reasonable that this is an independent prohibition. A partial measure has a rationale and a source (“any amount” of forbidden fat). So it is reasonable that this is the prohibition of cooking on the Sabbath. Consequently, it is also reasonable that he would be stoned for it, since one who cooks on the Sabbath is stoned.

When someone cooks unintentionally twice the amount that creates liability, he does not have to bring two sin-offerings, so the liability is for the act and not for the quantity. So why, when someone adds more than the required amount in a case of saving a life, would he be liable? That is why I thought maybe it is an independent prohibition, but as the Rabbi said, it has no source. Maybe one could say that what he cooked for saving a life is not considered cooking, because the Sabbath is permitted in life-threatening circumstances, and then adding to the quantity adds a new act of cooking? Although even if we say that the Sabbath is permitted, it is still clear that he cooked, just permissibly, so adding to the quantity did not add a new act of cooking.

Could it be that the Rabbi didn’t see the question?

I answered, and now I see that it doesn’t appear. Strange. I’ll write it again.
Your proof that this is a prohibition on the act is incorrect. The reason one is not liable for two offerings for double cooking is that this is one lapse of awareness. That is a rule in the laws governing the division of sin-offerings, not in the definition of the prohibition of cooking. Therefore one is liable only once even if this is a prohibition on the result. (By the way, in another article I argued that liability for a sin-offering is for the lapse of awareness, not for the transgression.)
On the face of it, the prohibition is not on the act but on the result. The one who becomes liable is the person who placed it on the fire (that is a condition for liability, not the prohibition itself). So there is no reason not to hold him liable for the added amount.
In our case only one prohibition was violated, because the first amount cooked for the sick person was a permitted act, so the division of sin-offerings is not relevant here. There is cooking of the second amount here, and therefore he is liable.

Thank you. And do those who disagree with the Ran hold as I wrote, that it is a prohibition on the act, or does everyone agree that the prohibition is on the result?

It is reasonable that they disagree about that, because otherwise there really is no reason it should not be a Torah prohibition, though one could force another reading.
In the reply that didn’t appear, I noted that there is room to connect this to the discussion among the later authorities about someone who placed food on the fire on the Sabbath and it finished cooking after the Sabbath (and similarly with sowing). Though that too can be rejected.