Shape the beef into 4 thin oval patties slightly larger than the bread and season them with the salt and pepper.
Melt 1 tablespoon butter on a griddle over medium heat and cook the onions for 12 to 15 minutes until deep golden; transfer them to a plate.
Cook the patties on the same griddle for 3 to 4 minutes per side until they reach 160°F.
Spread the outside of each bread slice with mayonnaise and build each sandwich with Swiss, one patty, onions and another slice of Swiss.
Melt the remaining butter and griddle the sandwiches over medium-low heat for 3 to 4 minutes per side until crisp and melted through.