01 - Melt butter and olive oil in a large skillet over medium heat. Add sliced onions, salt, and sugar if desired. Cook, stirring occasionally, for 15 to 20 minutes until golden and caramelized, reducing heat if browning too rapidly. Set aside.
02 - In a separate skillet over medium-high heat, melt butter and olive oil. Add mushrooms and cook 4 to 5 minutes until browned. Stir in garlic, salt, pepper, and thyme, cooking 1 to 2 minutes more. Remove from heat.
03 - Combine ground beef with salt, pepper, and Worcestershire sauce in a bowl. Gently mix and shape into 4 patties slightly larger than buns.
04 - Heat grill or skillet over medium-high heat. Cook patties 3 to 4 minutes per side for medium doneness. During the last minute, place a slice of Swiss cheese on each patty and cover to melt.
05 - Lightly toast buns on grill or in toaster until golden.
06 - Spread mayonnaise or aioli on bottom buns if desired. Add lettuce, burger patties, sautéed mushrooms, caramelized onions, and top with the bun crown. Serve immediately.