Heat a skillet over medium heat and melt 1 tablespoon of butter.
Add the sliced onions and sauté until caramelized, about 10 minutes.
Form the ground beef into two patties and season with salt and pepper.
Increase heat to medium-high and cook the patties for 4-5 minutes on each side.
Place a slice of cheese on each patty during the last minute of cooking to melt.
Remove the patties and onions from the skillet and set aside.
Butter one side of each slice of bread.
Assemble the sandwich with a patty, onions, and another slice of cheese.
Top with another slice of bread, buttered side up.
Grill the sandwich in the skillet until golden brown, about 3-4 minutes per side.