Despite appearances, or the way it may be worded, you are actually requested to have performed 30 incisions into corpses and harvested 30 pieces of flesh. This means you'll obtain this after having operated on at least your 30th corpse, which may take a bit in the game to come across depending on your carrot output. After a while, the Donkey will demand carrots for corpses.
The imagery suggests that you would be turning them into meat slices at the table in your house, but that is misleading. Just extract Flesh from corpses any time you can and this should pop quite early as you're working on the quests.