01 - In a large skillet over medium heat, melt butter with olive oil. Add sliced onions, salt, and sugar. Cook, stirring frequently, for 20-25 minutes until onions are deep golden and caramelized. Remove from heat and set aside.
02 - In the same skillet, melt butter over medium heat. Add sliced mushrooms and cook 6-8 minutes until moisture evaporates and mushrooms brown. Stir in garlic, salt, pepper, and thyme if using; sauté 1-2 minutes more. Remove and keep warm.
03 - Combine ground beef, Worcestershire sauce, salt, and pepper in a bowl. Gently mix and divide into four equal patties. Press a slight indent into the center of each patty to prevent puffing during cooking.
04 - Heat grill or skillet over medium-high heat. Cook patties 3-4 minutes per side for medium doneness or until desired level. Place Swiss cheese slices on patties during the last minute, cover to melt.
05 - Lightly toast brioche or hamburger buns on the grill or in a toaster until golden.
06 - Spread mayonnaise on the bottom buns if using. Layer with lettuce leaves, cooked patties topped with Swiss cheese, sautéed mushrooms, caramelized onions, and finish with top buns. Serve immediately.