01 - Melt butter and olive oil in a large skillet over medium heat. Add sliced onions, salt, and sugar if using. Cook, stirring occasionally, for 15-20 minutes until deep golden and caramelized. Adjust heat to avoid burning. Remove from heat and set aside.
02 - In a separate skillet over medium-high heat, melt butter and olive oil. Add sliced mushrooms and cook 4-5 minutes until browned. Stir in garlic, salt, pepper, and thyme if using; cook 1-2 additional minutes. Remove from heat.
03 - Combine ground beef, salt, black pepper, and Worcestershire sauce in a mixing bowl. Form into four equal patties slightly larger than the buns.
04 - Preheat grill or skillet to medium-high. Cook patties 3-4 minutes per side for medium doneness. In the final minute, place a slice of Swiss cheese atop each patty, cover to melt thoroughly.
05 - Lightly toast the split buns on the grill or in a toaster until golden.
06 - Spread mayonnaise or aioli on the bottom bun if desired. Add lettuce leaves, place the burger patty, then top with sautéed mushrooms and caramelized onions. Cover with the top bun.
07 - Serve immediately accompanied by preferred side dishes.