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.