01 - In a large skillet over medium heat, melt butter with olive oil. Add sliced onions and salt. Cook, stirring occasionally, until onions are deeply golden and caramelized, about 18–20 minutes. Add balsamic vinegar in the last minute. Remove from skillet and set aside.
02 - In the same skillet, melt butter with olive oil over medium-high heat. Add mushrooms and cook, stirring occasionally, until browned and their moisture has evaporated, about 6–8 minutes. Stir in garlic, salt, pepper, and thyme. Sauté for another 1–2 minutes. Remove from heat.
03 - Divide ground beef into 4 equal portions and gently shape into patties slightly larger than your buns. Season both sides with salt and pepper.
04 - Heat a grill, grill pan, or skillet over medium-high heat. Cook patties for 3–4 minutes per side for medium doneness, or until desired doneness. In the last minute, place a slice of Swiss cheese on each patty, cover, and let melt.
05 - Spread mayonnaise and mustard on the cut sides of toasted buns if desired. Place lettuce on the bottom bun, add the burger patty with melted Swiss cheese, top with sautéed mushrooms and caramelized onions. Finish with the top bun.
06 - Serve immediately with your favorite side dish.