Melt 1 tbsp butter in a large skillet over medium heat, add onion and a pinch of salt, and cook 15–20 minutes, stirring often, until deep golden. Set aside.

In a second skillet, melt remaining 2 tbsp butter over medium-high heat. Add mushrooms and cook undisturbed 2 minutes, then stir in garlic and cook 6–8 minutes until browned and tender.

Shape beef into four 1/2-inch-thick patties, press a small indent in centers, and season both sides with salt and pepper. Wipe out the onion skillet, add oil, and heat over medium-high. Cook patties 4–5 minutes per side.

In the last minute of cooking, top each patty with a slice of Swiss. Add a little water and cover to melt if needed. Toast buns cut-side down in the same pan about 1 minute.

Layer onions on bottom buns, add patties, heap mushrooms on top, and close with bun tops. Serve hot.