Form the patties: Divide the ground beef (80/20) into 4 equal portions (about 6 ounces each). Gently shape into 3/4-inch-thick patties and press a shallow 1-inch-wide dimple in the center of each to prevent doming.
Season: Sprinkle both sides of the patties evenly with the kosher salt and black pepper; set aside at room temperature while you cook the mushrooms.
Cook burgers: Return the skillet to medium-high. Add the patties and cook until well browned on the first side, 3–4 minutes. Flip, top each patty with a mound of the mushroom mixture and a slice of Swiss cheese, then cover the pan and cook until the cheese is melted and the centers read 130–135°F for medium-rare (or to your preferred doneness), 2–3 minutes more. Transfer to a plate to rest.
Toast buns: Place the hamburger buns cut-side down in the buttery skillet and toast until golden at the edges.
Assemble and serve: Spread the mayonnaise on the toasted buns. Set a cheesy mushroom-topped patty on each bottom bun, cap with the top bun, and serve hot.