In a large skillet, preferably cast iron, melt 2 tablespoons of butter over medium heat. Add onions and cook, stirring often, until they are soft, translucent, and starting to brown.
2 tablespoons butter, divided, 1 ½ medium sweet onion, thinly sliced
While the onions cook, make the burger patties in the shape of the bread. In a large bowl, combine the ground beef, Worcestershire, salt, and pepper. Divide into 2 portions.
½ pound ground beef, 1 teaspoon Worcestershire sauce, ½ teaspoon salt, ½ teaspoon ground black pepper
When the onions are done, remove them to a plate. Wipe the skillet clean with paper towels. Cook the patties in the same skillet over medium-high heat until browned and cooked through to your desired temperature. Remove the patties from the skillet and wipe the skillet clean.
In the same skillet, melt 3 tablespoons of butter over medium heat. Add one slice of bread, butter side down, sauce side up. Add 1 slice of cheese, the cooked hamburger patty, a heaping amount of caramelized onions, a second slice of cheese, and another slice of bread, sauce side down, buttered side up. Repeat with the second sandwich.
4 slices rye or sourdough bread , 8 slices Swiss cheese or Cheddar cheese
Cook until the bread is golden brown and the cheese is melted, about 3 to 4 minutes. Gently flip the sandwiches and cook until golden brown and the cheese is melted. Serve immediately.