01 - Melt butter with olive oil in a large skillet over medium heat. Add sliced onions, salt, and sugar. Cook, stirring frequently, for 20-25 minutes until deeply golden. Transfer to a bowl and set aside.
02 - In the same skillet, melt butter over medium heat. Add mushrooms and cook 6-8 minutes until browned and moisture evaporates. Stir in garlic, salt, pepper, and thyme (if using), sauté an additional 1-2 minutes. Set aside.
03 - Combine ground beef with Worcestershire sauce, kosher salt, and black pepper gently. Divide into 4 equal portions, pressing a slight indent in the center of each patty.
04 - Heat grill or skillet to medium-high. Cook patties for 3-4 minutes per side for medium doneness. In the last minute, top each with Swiss cheese and cover to melt.
05 - Lightly toast split buns on the grill or toaster until golden.
06 - Spread mayonnaise on bottom buns if using. Layer lettuce, burger patties, sautéed mushrooms, caramelized onions, and top with bun halves. Serve immediately.