01 - In a large skillet over medium-low heat, melt butter and olive oil. Add sliced onions and salt. Cook, stirring occasionally, for 20-25 minutes until deep golden brown and soft. Add sugar if desired for extra sweetness. Remove from pan and set aside.
02 - In the same skillet, heat butter and olive oil over medium heat. Add mushrooms, salt, pepper, and thyme. Cook for 6-8 minutes until mushrooms are browned and tender. Set aside.
03 - Divide ground beef into 4 equal portions and shape into patties slightly larger than your buns. Season both sides with salt and pepper.
04 - Preheat a grill or skillet over medium-high heat. Cook patties for 3-4 minutes per side for medium doneness, or until desired doneness is reached. In the last minute of cooking, place a slice of Swiss cheese on each patty, cover, and let melt.
05 - Spread butter on the cut sides of each bun. Toast on the grill or in a pan until golden brown.
06 - Spread mayonnaise and Dijon mustard on the bottom bun. Add lettuce and tomato. Place the burger patty with melted Swiss cheese on top, then add a generous spoonful of sautéed mushrooms and caramelized onions. Cap with the top bun and serve immediately.