Mustard-fried burger patty
Here is Kenji Alt's in-depth recreation of the double-double animal-style burger.  The core of his technique is:


  The process is simple: Sear the patty on one side, and squirt some
  mustard on it as it sizzles. Flip the patty over so that the mustard
  cooks into the second side.
  
  The patties are covered with the cheese, then the caramelized onions
  are applied liberally to a single patty before topping it with the
  second, fusing all the elements together into a single cheesy, beefy, sweet, oniony, gooey, salty, oozy, crispy, meaty, savory, melty, delicious mess.


He uses basic yellow mustard, btw.

In response to the question from comments about how long to cook, here is what the recipe says:


  Add burger patties and cook without moving until well browned and
  crusty on first side, about 2 1/2 minutes. While they are cooking,
  spread 1 tablespoon mustard on raw side of each patty with a spoon.
  Meanwhile, top each bottom bun with up half of spread, 4 slices
  pickles, 1 slice tomato, and lettuce. Flip patties with a thin spatula
  so mustard side is down and continue to cook for 1 minute