1. In a mixing bowl, combine all the ingredients for the special sauce. Set aside.
2. Heat a skillet over medium heat. Add butter and sliced onions. Cook until onions are caramelized.
3. Form ground beef into 4 patties. Season with salt and pepper.
4. In the same skillet, cook patties until desired doneness.
5. Assemble the patty melts: Spread special sauce on rye bread slices, add a patty, caramelized onions, and a slice of Swiss cheese. Top with another slice of bread.
6. Grill the sandwiches in the skillet until the bread is golden brown and the cheese is melted.