Prep the Onions: Heat 2 tablespoons of butter in a large skillet over medium heat. Add the sliced onions and cook, stirring now and then, until they are golden brown and caramelized. This takes about 15-20 minutes. Remove from the skillet and set aside.

Form the Patties: While the onions are cooking, divide the ground beef into 4 equal portions and shape them into thin patties. Season each patty with salt, pepper, and optionally, a splash of Worcestershire sauce.

Cook the Patties: In the same skillet, cook the beef patties over medium-high heat for about 3-4 minutes on each side, or until they are cooked the way you like. Remove from the skillet and set aside.

Assemble the Sandwiches: Butter one side of each slice of bread. Place a slice of cheese on the unbuttered side, then a beef patty, caramelized onions, another slice of cheese, and top with another slice of bread, buttered side out.

Grill the Sandwiches: Heat the skillet or griddle over medium heat. Cook the sandwiches for about 3-4 minutes on each side, or until the bread is golden brown and the cheese is melted.

Serve: Remove the sandwiches from the skillet and let them rest for a minute before cutting them in half and serving.