In a large skillet, melt 2 tablespoons butter over medium heat. Add the chopped onion, water, and ¼ teaspoon of salt. Cover and cook, until tender, about 5 minutes.
12 tablespoons salted butter, 1 medium sweet onion, 1 tablespoon water, ¾ teaspoon kosher salt
Reduce the heat to medium-low and remove the lid. Continue to cook until onions are translucent and just beginning to brown, about 3 more minutes. Transfer the 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 4½x½-inch patty. Sprinkle both sides of the patties with salt and pepper. Refrigerate the patties for 15-30 minutes.
1 pound 90% lean ground beef, ¾ teaspoon ground black pepper
Spread each bun top with 2 tablespoons of butter; set aside.
4 homemade burger buns
Heat 1-2 tablespoons butter in the same skillet used for the onions until it begins to smoke. Use a spatula to transfer the patties to the skillet and cook for 3 minutes (without moving them).
Flip the patties and cook for 1 more minute. Place 1 cheese slice over each patty and continue to cook until the cheese is melted, about 20-30 seconds more.
4 slices American cheese
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.