01 - Heat olive oil and butter in a large skillet over medium-low heat. Add onions and sprinkle with salt and sugar. Cook, stirring occasionally, until golden and caramelized, about 20-25 minutes. Remove from skillet and set aside.
02 - In the same skillet, heat olive oil and butter over medium heat. Add mushrooms and cook until softened and golden, about 6-8 minutes. Stir in garlic, season with salt and pepper, cook for 1 minute, then set aside.
03 - Shape the ground beef into 4 equal patties, about ¾ inch thick. Season both sides with salt, pepper, and Worcestershire sauce.
04 - Heat a grill or skillet over medium-high heat. Cook patties for 3-4 minutes per side for medium doneness. Place cheese slices on top during the last minute of cooking; cover to melt.
05 - Toast burger buns until golden and crisp.
06 - Spread mayonnaise and mustard on the bottom bun if using. Add lettuce, then the beef patty with melted cheese, followed by caramelized onions and sautéed mushrooms. Top with tomato if desired and the top bun.
07 - Serve the burgers immediately while hot and the cheese is melted.