01 - Heat butter and olive oil in a large skillet over medium-low heat. Add sliced onions, salt, and sugar. Cook, stirring occasionally, for 18-20 minutes until onions are golden brown and caramelized. Remove from skillet and set aside.
02 - In the same skillet, heat butter and olive oil over medium-high heat. Add sliced mushrooms, salt, pepper, and thyme if using. Sauté for 6-8 minutes until mushrooms are browned and excess liquid evaporates. Remove and set aside.
03 - Divide ground beef into 4 equal portions and shape into patties slightly larger than the buns. Season both sides evenly with salt and pepper.
04 - Preheat grill or skillet to medium-high heat. Cook patties 3-4 minutes per side for medium doneness or to preference. During the last minute, top each patty with Swiss cheese and cover to melt.
05 - Optionally spread mayonnaise on bottom buns. Layer arugula or lettuce, followed by the cheeseburger patty, then spoon sautéed mushrooms and caramelized onions on top. Finish with the top bun.
06 - Serve the assembled burgers immediately while hot.