Add 1 tablespoon butter to a large saute pan over medium heat. Add onions and cook, stirring occasionally, until browned and slightly caramelized (about 15 minutes). Set aside.
Form ground beef into 4 thin oval shaped burger patties (similar to the shape of the bread). Season on each side with salt and pepper.
Add 1 tablespoon butter to the saute pan and turn heat to medium-high. Add burgers and cook on each side for 2-3 minutes or until the burgers are cooked to your desired doneness.
Assemble the sandwiches by layering one slice of bread, one slice of cheese, one burger patty, 1/4 of the onions, another slice of cheese, and another slice of bread. Assemble remaining sandwiches with the rest of the ingredients.
Add two sandwiches at a time and cook on the first side for 2-3 minutes or until a dark golden brown crust forms and the cheese is melted. Gently press the sandwich with spatula to smoosh everything together and toast the bread evenly.
Carefully remove sandwiches from the pan and melt another 1/2 tablespoon of butter in the pan. Add the two sandwiches back to the pan, uncooked side down, and cook for another 2-3 minutes or until a dark golden brown crust forms on the second side and the cheese is melted. Remember to gently press the sandwich with a spatula.
Continue cooking the other two sandwiches in the same way with the remaining tablespoon of butter.
Slice, serve, and enjoy!
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