01 - In a large skillet over medium heat, melt butter and olive oil together. Add sliced onions, salt, and sugar. Cook, stirring occasionally, until onions are deeply golden and soft, about 15 to 20 minutes. Stir in balsamic vinegar if using. Remove from heat and set aside.
02 - Divide ground beef into four equal portions and shape into patties approximately 3/4 inch thick. Season both sides with salt and freshly ground black pepper.
03 - Preheat grill or skillet over medium-high heat. Cook patties for 3 to 4 minutes per side for medium doneness. During the final minute, place a slice of cheddar on each patty and allow to melt.
04 - Lightly toast burger buns on the grill or in a toaster until golden.
05 - Spread mayonnaise and Dijon mustard on the bottom halves of the buns. Layer with lettuce, tomato slices, the melted cheese-topped patty, caramelized onions, and pickles if desired. Cap with the top half of the bun.
06 - Present immediately alongside preferred accompaniments.