Preheat the oven to 250 degrees F.
Put the bottoms of the buns on a baking sheet. Put in the oven to keep warm while you make the burgers.
Divide the beef into 4 equal patties, 3 3/4- to 4-inches in diameter and 1/2-inch thick (the burgers will be bigger than the buns). Sprinkle the patties with 1 teaspoon salt and 1/2 teaspoon pepper.
Heat the butter in a large skillet or on a griddle over medium-high heat. When the foam subsides and the butter is just beginning to brown, add the patties and cook until well browned, about 3 minutes. Flip and top each burger with 1 half of American cheese. Place 2 strips of Swiss cheese on either side of the American. Trim the cheese to just fit the patty. Continue to cook until browned on the outside but still slightly pink in the center, 2 to 3 minutes more.
Place a patty cheese-side down on a bottom bun and, if using, add the grilled onions. Press the top over the burger. Repeat for the remaining patties. They will be juicy and oozy. Serve on a piece of wax paper with potato chips on the side.