Melt 2 tablespoons of butter in a cast-iron or non-stick skillet over medium heat. Add onions, season with salt and pepper, and stir to combine.
Cook onions slowly for at least 20 to 25 minutes, stirring occasionally to prevent burning. You may need to adjust your heat to low as well. Cook onions until they become translucent and golden brown and then allow to cool.
Cream butter in small bowl using a fork or electric mixer.
Gradually blend in caramelized onions. Butter mixture can be used while still soft; or it can be rolled into a cylinder in parchment paper or plastic wrap and refrigerated or frozen, then sliced into thin rounds.
Flavored butter can be stored a couple days in the refrigerator or it can be frozen for several weeks.
Divide meat into 6 equal portions (about ⅔-cup each) and flatten portions into 4½-inch round patties.
Broil or grill patties.
Place 1 Tablespoon of caramelized onion butter on each patty and top with a slice of cheddar cheese.
Serve on rolls with optional lettuce, tomatoes, and bacon.