Form ground beef into 4 patties, approximately 1/2 inch thick. Set aside.
Melt 4 Tablespoons butter in large non-stick skillet over medium-high heat.
Add onions. Season with 1 teaspoon salt and 2 teaspoons black pepper. Saute until brown, approximately 8-10 minutes.
Remove onions from pan and set aside.
In same pan used to cook onions, place hamburger patties. Season with remaining salt and pepper. Cook until browned on one side, approximately 5-6 minutes. Flip and cook second side an additional 4-5 minutes. Remove and set aside.
Reduce heat to medium. Spread 1/4 teaspoon of butter on one side of all the slices of bread. On opposite sides, spread mustard, evenly. Lay four slices of bread in pan, butter side down.
Top each slice of bread with one slice cheese, burger patty and a 1/4 of the onion mixture.
Add second slice of cheese to sandwich, then top with second slice of bread, butter side up.
Grill the sandwiches for 3-4 minutes on each side until the bread is golden brown and the cheese has melted.