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.
Now carefully move each loaded hamburger patty to the slices of bread with the mustard on top.
Now place 2 tablespoons of butter on the griddle and spread it around to cover the surface. Place the stacked patty melts on the griddle, and cover with the additional slices of bread.
Toast the bread for 2 to 3 minutes, until golden brown. Turn the heat down to medium-low if needed and watch them carefully.
Once they are ready to flip, add the final 2 tablespoons of butter to the griddle. Flip the patty melts over as the butter melts. Toast for another 2 to 3 minutes to make sure the second side is perfectly toasted.
Flip the patty melts onto a cutting board, making sure the cheese is on top of the patties. Cut them in half and serve immediately.