In a skillet over medium heat, melt 1 tablespoon of butter. Add the thinly sliced onions and sauté until they become soft and caramelized, about 8-10 minutes. Season with salt and pepper to taste. Remove from heat and set aside.

While the onions are cooking, divide the ground beef into four equal portions and shape them into patties. Season each side with salt and pepper.

In the same skillet over medium-high heat, add a bit more butter. Once melted, add the beef patties and cook for about 3-4 minutes per side, or until they reach your desired level of doneness. During the last minute of cooking, place a slice of American cheese on top of each patty to melt.

In a separate skillet or griddle, melt the remaining butter. Place two slices of bread in the skillet, add a patty with melted cheese, a generous portion of caramelized onions, and then top with another slice of bread. Grill until both sides are beautifully golden brown.

Remove from heat, slice in half, and serve immediately, optionally adding your favorite condiments.