Why not lend an old friend a hand? I'm sure it would have been his last wish.
Collect 10 brains.
Head back to the town of Hallow's End and go on a zombie killing rampage. Zombies will only drop their brains if you kill them with a headshot/critical hit. Return to TK after collecting 10 brains.
"You gave him a few brains. He appears to be coughing up an item... "
Wait for a few seconds after talking to T.K. He will eat a brain then vomit up an item. It will be a Shock Artifact if not already received; otherwise, he drops a random item.
Any brains picked up without this mission or the ones following do not count and are unusable.
Keep in mind that as long as you stay in a zone the brains that drop from zombies will remain on the ground until you pick them up. Therefore, if you have completed the mission leave the extra brains on the ground so you can collect them when you get the next mission.
A great tip for "brain hunting" is to simply walk all around the deserted town in front of T.K.'s house and kill zombies for the first 2 brains missions. After that, just run back and forth between Zombie T.K.'s house and the entrance to Hallow's End. Be sure to actually leave Hallow's End if you're having trouble finding zombies, then return in a few seconds to see if some spawn. Make sure you pick up the zombie brains before leaving, as they will disappear upon exiting Hallow's End.
When T.K. is talking about all the different ways to cook brains, this is from the movie 'Forrest Gump' in which the title character's friend lists all the ways to prepare shrimp that he can think of. See: Benjamin Buford 'Bubba' Blue of Forrest Gump
T.K.'s quote about brains, that you can "Boil 'em, mash 'em, stick 'em in a stew..." is a reference to The Lord of the Rings: The Two Towers in which Samwise Gamgee explains to Gollum what "taters" are.
Despite the background, T.K. does not need to be dead to begin this.