Shape beef into 4 thin patties slightly larger than slices of bread. Heat large cast iron skillet on medium-high. Season patties with ½ teaspoon each salt and pepper and cook 2 minutes per side; transfer to a plate.
Wipe out skillet and return to medium-low heat. Spread mayo on 1 side of each slice of bread. In 2 batches, place 2 slices bread mayo-side down in the skillet.
Top each with 1½ slices Swiss cheese, a beef patty, caramelized onions, 1½ slices American cheese, and second slice of bread, mayo-side up.
Place small skillet (wrapped in foil) on top of sandwiches to press and cook until bread is golden brown and crisp, 3 minutes per side.