Preheat an outdoor grill for medium-high heat, and lightly oil the grate.
Divide the beef in half, and form each half into a patty. Grill the beef patties for 8 to 10 minutes per side, or to desired doneness.
While the burgers are grilling, melt 1/4 cup of butter in a large skillet. Lay out 4 slices of bread, and top each slice with 2 slices of American cheese. Top each sandwich with a slice of bread to make 4 cheese sandwiches. Pan-fry the sandwiches in the butter until golden brown, about 3 minutes per side. Set the sandwiches aside.
Melt the remaining 2 tablespoons of butter in the skillet, and cook the sliced onions until browned, stirring occasionally, about 8 minutes.
To assemble, top each grilled burger with half the onions, and place each between 2 grilled cheese sandwiches.