1. In a skillet over medium heat, melt 1 tablespoon of butter. Add the sliced onions and cook until golden brown, about 10 minutes. Set aside.
2. Form the ground beef into two equal patties and season with salt and pepper.
3. In the same skillet, cook the patties for about 4-5 minutes on each side or until cooked to your liking.
4. During the last minute of cooking, place a slice of cheddar cheese on each patty to melt.
5. Remove the patties from the skillet and set aside.
6. Butter one side of each slice of rye bread. Place two slices in the skillet, buttered side down.
7. Top each slice with a patty, caramelized onions, and another slice of bread (buttered side up).
8. Grill until golden brown on both sides and the cheese is melted, about 3-4 minutes per side.
9. Slice in half and serve hot.