In a large skillet over medium heat, melt 1 tablespoon of butter. Add the thinly sliced onions and sauté until golden brown and caramelized, about 10-12 minutes. Transfer to a plate and set aside.
Increase the heat to medium-high. In a mixing bowl, combine the ground beef with Worcestershire sauce, salt, and pepper. Divide into two equal portions and shape into patties.
Cook the beef patties in the hot skillet for 3-4 minutes on each side, until preferred doneness. During the last minute, place one slice of Swiss cheese on each patty and cover with a lid to melt the cheese.
Spread butter on one side of each slice of rye bread. Remove the cooked patties from the skillet and set aside.
In the same skillet, lay down one slice of rye bread, buttered side down. Place a cheese-topped patty on this slice, add caramelized onions, and top with a second slice of Swiss cheese. Add the second slice of rye bread on top, buttered side up.
Reduce heat to medium-low and grill the sandwich for 3-4 minutes on each side until golden brown and the cheese is melted. Press down with a spatula for even toasting. Repeat for the second patty melt.
Once both sandwiches are golden and heated through, remove from skillet and let cool for a minute. Slice each sandwich diagonally in half for presentation.
Notes
Serve with crispy fries or a light salad for a complete meal.