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.