01 - Heat butter and olive oil in a large skillet over medium-low heat. Add sliced onions along with salt and sugar, and cook, stirring occasionally, for 18 to 20 minutes until golden and caramelized. Remove from heat and set aside.
02 - In the same skillet, heat butter and olive oil over medium-high heat. Add sliced mushrooms, salt, black pepper, and thyme if using. Sauté for 6 to 8 minutes until mushrooms are browned and moisture has evaporated. Remove from heat.
03 - Divide ground beef into four equal portions and shape each into a patty slightly larger than the buns. Season both sides with salt and black pepper.
04 - Preheat grill or skillet to medium-high heat. Cook patties 3 to 4 minutes per side for medium doneness or to preference. During the final minute of cooking, place one slice of Swiss cheese on each patty and cover to melt the cheese.
05 - If desired, spread mayonnaise on the bottom halves of the toasted buns. Layer arugula or lettuce leaves, then the cheeseburger patties, followed by a generous amount of sautéed mushrooms and caramelized onions. Top with the other bun halves.
06 - Serve immediately while hot for the best flavor and texture.