Melt 1 tablespoon butter in a skillet over medium-low heat and cook the onions for 18 to 20 minutes until deep golden.
Shape the beef into 4 thin oval patties slightly larger than the bread and season with salt and pepper.
Cook the patties on a hot griddle for 3 to 4 minutes per side without pressing until they reach 160°F.
Butter one side of each rye slice with the remaining butter and lay 4 slices butter-side down on a board.
Top each with Swiss, one patty, browned onions, another Swiss slice and a mustard-spread rye slice with its buttered side facing out.
Griddle the melts over medium-low heat for 3 to 4 minutes per side until the bread is crisp and the cheese is melted.
Rest the melts for 1 minute, then cut each diagonally and serve.
