01 - Melt butter with olive oil in a large skillet over medium heat. Add sliced onions, salt, and sugar. Cook, stirring occasionally, for 15 to 20 minutes until onions are soft and golden brown. Stir in balsamic vinegar if desired, then remove from heat and set aside.
02 - Form ground beef into four equal patties, approximately 3/4 inch thick. Season both sides with salt and black pepper.
03 - Preheat a grill or skillet to medium-high heat. Cook patties for 3 to 4 minutes per side for medium doneness. Place a slice of cheddar on each patty during the last minute to melt.
04 - Lightly toast the split burger buns on the grill or in a toaster until golden.
05 - Spread mayonnaise and Dijon mustard on the bottom half of each bun. Layer with lettuce, tomato slices, cooked patty with melted cheese, caramelized onions, and pickles if desired. Cover with the top half of the bun.
06 - Serve immediately with preferred side dishes.