Set out 12 slices of thick-cut sandwich bread. Slather a generous teaspoon of yellow mustard over six of the slices.
Preheat a griddle, or a large skillet, to medium heat. (If using a skillet, you will most likely have to work in batches.) Carefully move the patties to the griddle or skillet. Sear the patties for 3 minutes per side.
Once you have flipped the patties over, place an equal portion of caramelized onions on top of each patty, spreading them out so the onions cover the meat entirely. Layer one slice of American cheese and one slice of Swiss cheese on top of each patty. Continue to cook until the hamburger patty has cooked for at least 6 minutes, total.
