01 - In a large skillet over medium heat, melt butter with olive oil. Add sliced onions, sprinkle with salt and sugar, then cook stirring frequently for 20-25 minutes until 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 they release moisture and start browning. Add garlic, salt, pepper, and thyme if using; sauté for 1-2 minutes more. Set aside.
03 - Gently combine ground beef with Worcestershire sauce, salt, and pepper in a bowl. Divide into 4 equal patties, making a slight indentation in the center of each.
04 - Heat grill or skillet over medium-high heat. Cook patties 3-4 minutes per side for medium doneness. In the final minute, top each patty with Swiss cheese slice and cover to melt.
05 - Lightly toast bun halves on the grill or in a toaster until golden.
06 - Spread mayonnaise on bottom buns if using, add lettuce if desired, then place patties, sautéed mushrooms, caramelized onions, and top with bun halves. Serve immediately.