01 - In a large skillet over medium heat, melt the butter. Add sliced onions, sugar, and a pinch of salt. Cook, stirring occasionally, for 15–20 minutes or until onions are golden brown and soft. Set aside.
02 - In another skillet, heat olive oil over medium-high heat. Add mushrooms and cook for 5–7 minutes until browned and tender. Season with salt, pepper, and thyme. Remove from heat.
03 - In a bowl, gently mix ground beef with salt and pepper. Form into 4 equal patties, pressing a small indentation in the center of each to prevent puffing during cooking.
04 - Heat a grill or skillet over medium-high heat. Cook the patties for 3–4 minutes per side for medium doneness. In the last minute of cooking, top each patty with a slice of Swiss cheese and cover to melt.
05 - Lightly toast brioche buns on the grill or in a toaster.
06 - Spread mayonnaise on the bottom buns. Add arugula or lettuce, then place the burger patties on top. Pile on the sautéed mushrooms and caramelized onions. Top with the upper buns and serve immediately.