"Kill chef" is an objective in the quest The Hunger Pangs in The Horrible Hunger of the Ravenous Wattle Gobbler DLC
After Torgue announces that your new item on the Wattle Gobbler's menu is murder then the chefs turn on you. Might have something to do with Torgue saying the Vault Hunter is there to kill all the Chefs too.
The chefs are a mixed bunch of reskinned Marauders, Psychos and Brutes. There are still a number of unique foes that you will need to face and kill to get through here.
There are 2 unique chefs that you will encounter: Chef Gouda Remsay, who spawns from the Southwestern corner of the area. He is the head chef of the kitchen. The Sous chef is a Midget Rat: The Rat in the Hat
After you kill Chef Gouda Remsay, you will be contacted by Anton Smith, president of the Torgue Corporation. He tells you to stop listening to Mr Torgue. Once he finishes talking the objective will update.