Preheat a skillet over medium heat.
Melt 1 tablespoon of butter in the skillet.
Add the sliced onions and cook until caramelized, about 8-10 minutes.
Season the onions with salt and pepper, then remove from the skillet.
Form the ground beef into two patties and season with salt and pepper.
Cook the patties in the skillet for about 4-5 minutes on each side, until cooked through.
Place a slice of cheese on each patty during the last minute of cooking to melt.
Butter the outside of the bread slices.
Place one slice of bread in the skillet, add a patty with cheese, top with caramelized onions, and place another slice of bread on top.
Grill the sandwich until golden brown, then flip and repeat with the other side.