There’s a dispute between רבי and ר יוסי regarding the laws of לישה- kneading on Shabbos. Rebbe holds the melacha is even with just mixing the flour and water. R Yosi holds the melacha is only when one manually kneads the two. R Yosi simply holds that manual kneading is the forbidden act and it is true even with materials that mix well by themselves, like ashes with water or dyes.

Acc to רבה- even Beis Hillel agrees that one is not allowed to have a mill grind wheat by itself on Shabbos bc it makes loud noise and it disgraces the Shabbos. Acc to רב יוסף there is no such problem and only Beis Shammai wouldn’t allow the mill.

Regarding the dispute Beis Shmmai and Hillel about vessels resting on Shabbos, it is only about vessels doing work. But even Beis Shammai would allow incense to burn on ones property (soil) on Shabbos bc it is not a vessel doing the work.

Even Beis Shammai allows ones lamp to burn on Shabbos bc it they make the lamps הפקר so it’s not their vessels doing work.

Even beis Hillel agrees that food can’t be cooking by itself on Shabbos bc we’re concerned a person may stoke the coals cooking the food to expedite the process. We’re not concerned by foods where opening the oven would harm the food bc a person wouldn’t stoke the coals and harm the food. We’re also not concerned in cases where the oven is sealed bc there is a built in reminder not to stoke the coals since it is a bother to remove the seal.

Even for foods that are forbidden to be cooked by themselves on Shabbos, it is permitted by קדרא חייתא - if the stuff is totally raw bc then we aren’t scared someone would stoke the coals bc anyways the food isn’t for Friday night and by day it will get cooked by itself.

One cannot stir a pot of dye on the fire. (Tosafos- bc of dying and Rashi- bc it expedites the cooking process)