Divide ground beef into four patties. Heat 12-inch skillet over medium heat. Melt 1 tablespoon of butter in skillet and add two of the beef patties. Season each beef patty with garlic salt and Worcestershire sauce. Cook until beef patty has browned on both sides, flipping once after about 5 minutes. Remove cooked beef patties from skillet and onto a plate. Repeat with remaining patties.
Wipe skillet clean with paper towels. Spread softened butter onto one side of bread and place onto skillet with the butter side down. Top bread with caramelized onions, then beef patty, and then with cheeses. Top with another slice of bread and spread the top with softened butter. Cooking two at a time, flip the patty melts once one side of the sandwich has reached a golden brown and toast the other, about 2 minutes. Cover with a lid to allow the cheese to melt more quickly, if needed.
Remove and serve immediately.
Top each of 6 bread slices with some of the onions, a cheese slice, and a burger patty. Top each burger with a cheese slice and a piece of bread. Using a table knife, spread butter over the top and bottom of each sandwich.
Heat a 12" nonstick skillet over medium heat. Working in 3 batches, cook sandwiches, flipping once, until golden brown and hot, about 6 minutes.