Heat up a small pan or skillet over medium-low heat. Melt 1 tablespoon of butter, then add the sliced yellow onion. Turn the heat down on low and cook the onions, stirring continuously. This will carmelize them (instead of frying them). Cook for 30 minutes.
During your quick stir breaks, split the ground beef into four and shape hamburger patties. Season both sides with the salt and pepper.
When the onions are finished, warm a cast-iron skillet over medium-high heat. Once hot, melt the other 1 tablespoon of butter and add the burger patties. Cook on each side for 5 minutes, then remove to a plate.
For each sandwich, add a slice of Swiss, the burger patty, a slice of Velveeta, then some caramelized onions to a slice of bread. Top the sandwich another slice, then spread some mayonnaise on that slice.
Drain the grease from the burgers into a can. Heat the skillet back up and place the sandwich on the skillet, mayonnaise side down. Spread mayonnaise on the top slice. Press the sandwich down with a spatula occasionally while cooking to flatten it out.
Flip the sandwich after 3 or 4 minutes, or until the bread is nice and toasty. When the cheese has melted and the bread is toasted, remove to a plate and serve.