01 - In a large skillet over medium heat, melt butter with olive oil. Add sliced onions, salt, and sugar. Stir frequently and cook for 20–25 minutes until onions turn deeply golden and caramelized. Transfer to a bowl and set aside.
02 - In the same skillet, melt butter over medium heat. Add mushrooms and cook for 6–8 minutes until moisture evaporates and they begin to brown. Add garlic, salt, pepper, and thyme if desired; sauté for another 1–2 minutes. Remove from heat.
03 - Combine ground beef, Worcestershire sauce, salt, and pepper gently in a bowl. Divide into 4 equal portions and shape into patties with a slight indent in the center of each.
04 - Preheat grill or skillet over medium-high heat. Cook patties for 3–4 minutes per side for medium doneness. During the last minute, place a slice of Swiss cheese on each patty and cover until melted.
05 - Lightly toast the split buns on the grill or in a toaster until golden and crisp.
06 - Optionally spread mayonnaise on bottom buns, add lettuce leaves if desired, then top with cheeseburger patties, sautéed mushrooms, caramelized onions, and cover with top buns. Serve immediately.