In a skillet over medium heat, melt 1 tablespoon of butter. Add the sliced onions and cook until caramelized, about 10-15 minutes, stirring occasionally.
Season the onions with salt and pepper, then add Worcestershire sauce and garlic powder. Mix well and set aside.
In the same skillet, form the ground beef into two equal patties, seasoning both sides with salt and pepper.
Cook the patties for about 4-5 minutes on each side, or until desired doneness. Place a slice of cheese on each patty during the last minute of cooking to melt.
Heat another skillet over medium heat and butter one side of each slice of rye bread.
Place two slices of bread, buttered side down, in the skillet. Top each with a cheesy patty and caramelized onions.
Place the remaining slices of bread on top, buttered side up, and grill the sandwiches for about 3-4 minutes on each side until golden brown and crispy.
Remove from the skillet, slice in half, and serve hot.