01 - Heat butter and olive oil in a large skillet over medium-low heat. Add sliced onions and cook, stirring frequently, for 15 to 20 minutes until soft and golden brown. Sprinkle sugar and salt halfway through cooking to enhance caramelization. Remove and set aside.
02 - Combine ground beef, salt, pepper, Worcestershire sauce, and Dijon mustard in a mixing bowl. Mix gently until just combined. Divide into eight equal portions and shape into patties slightly larger than the buns.
03 - Heat a grill pan or skillet over medium-high heat. Cook patties 2 to 3 minutes per side until desired doneness. In the final minute, place a slice of cheese on each patty and cover to melt.
04 - Lightly toast buns, cut side down, in a dry skillet or on the grill until golden brown.
05 - Spread mayonnaise and mustard on the bottom bun if desired. Add lettuce or pickle, then the cheesy patty, and top with caramelized onions. Cover with the top bun.
06 - Serve immediately while warm.