In a large skillet, melt 2 tablespoons butter over medium heat. Add chopped onion, water and ¼ teaspoon salt. Cover and cook, until tender, about 5 minutes. Reduce heat to medium-low and remove lid. Continue to cook until onions are translucent and just beginning to brown, about 3 more minutes. Transfer onions to a plate, and divide into 4 equal portions; set onions aside until ready to use.
Divide beef into 4 equal sized portions and gently shape each into a 4½x½-inch patty. Sprinkle both sides of the patties with salt and pepper. Refrigerate the patties 15-30 minutes.
Spread each bun top with 2 tablespoons butter; set aside.
Heat oil in the same skillet used for the onions until the oil begins to smoke. Use a spatula to transfer the patties to the skillet and cook 3 minutes (without moving them). Flip the patties and cook 1 more minute. Place 1 cheese slice over each patty and continue to cook until cheese is melted, about 20-30 seconds more.
Transfer patties to the bun bottoms and place ¼ of the sautéed onions on each patty. Top each with the buttered bun tops and serve immediately.
1. Be sure to cook the onions in butter and water and then brown them.2. Don’t skimp on buttering the bun tops. Yes, 2 tablespoons per bun top to be considered Wisconsin Butter Burgers.3. Be sure to cook the patties a full 3 minutes before flipping them. These burgers need a good sear to get the flavorful dark brown crust.