Step 1

Melt 1 tablespoon butter in medium saucepan over medium heat. Add onion, water, and 1/4 teaspoon salt and cook, covered, until tender, about 5 minutes. Remove lid and continue to cook until translucent and just beginning to brown, about 3 minutes. Cover and keep warm.

Step 2

Transfer beef to rimmed baking sheet and separate into 4 equal mounds. Gently shape each mound into 4 1/2-inch-wide by 1/2-inch-thick patty. Combine 3/4 teaspoon salt and 3/4 teaspoon pepper in bowl and sprinkle both sides of patties with mixture. Refrigerate until ready to cook, up to 30 minutes.

Step 3

Spread 2 tablespoons butter onto each bun top; set aside. Heat oil in 12-inch skillet over high heat until just smoking. Using spatula, transfer patties to skillet and cook without moving them for 3 minutes. Flip patties and cook for 1 minute. Top each burger with1 slice of American cheese and continue to cook until cheese is melted, about 30 seconds longer.

Step 4

Transfer burgers to bun bottoms. Divide onion mixture among burgers and cover with buttered bun tops. Serve immediately.