01 - In a large skillet over medium heat, melt butter with olive oil. Add sliced onions and sprinkle with salt and sugar. Cook, stirring frequently, for 20 to 25 minutes until onions are deeply golden and caramelized. Transfer to a bowl and set aside.
02 - Using the same skillet, melt butter over medium heat. Add mushrooms and cook until they release moisture and begin to brown, about 6 to 8 minutes. Add garlic, salt, pepper, and thyme if using; sauté for 1 to 2 minutes more. Set aside.
03 - In a bowl, gently mix ground beef with Worcestershire sauce, salt, and pepper. Divide mixture into 4 equal portions and shape into patties with a slight indent in the center of each.
04 - Preheat a grill or skillet over medium-high heat. Cook patties for 3 to 4 minutes per side for medium doneness or until preferred level is reached. During the last minute, place a slice of Swiss cheese on each patty and cover until melted.
05 - Lightly toast the bun halves on the grill or in a toaster until golden.
06 - Spread mayonnaise, if desired, on the bottom buns. Layer with lettuce leaves, the cooked patties topped with melted cheese, sautéed mushrooms, and caramelized onions. Cap with top buns and serve immediately.