In a large skillet, melt 2 tablespoons butter over medium heat. Add chopped onion, water and 1/4 teaspoon salt. Cover and cook, until tender, about 5 minutes. Reduce heat to medium-low and remove lid. Continue cooking 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 aside until ready to use.
Divide beef into 4 equal sized portions and gently shape each into a 41/2" x 1/2" patty. Sprinkle both sides of the patties with salt and pepper; refrigerate the patties 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 one cheese slice on each patty and continue to cook until cheese is melted, about 30 seconds.
Transfer patties to the bun bottoms and place 1/4 of the saut�ed onions on each patty. Top each with the buttered bun tops and serve immediately.