Step 1: Prepare the Onions. In a large skillet over medium heat, melt 2 tablespoons of softened butter. Add the chopped sweet onions and 1 tablespoon of water. Sprinkle with ½ teaspoon of kosher salt. Sauté the onions, stirring occasionally, until they are caramelized and soft, about 15-20 minutes. Remove from heat and set aside.
Step 2: Make the Burger Patties. In a bowl, combine the ground beef with 2 tablespoons of softened butter, ½ teaspoon of kosher salt, and the ground black pepper. Use your hands to mix gently until just combined, being careful not to overwork the meat. Divide the mixture into four equal portions and shape them into patties.
Step 3: Cook the Burgers. In the same skillet used for the onions, heat over medium-high heat. Once hot, add the burger patties. Cook for about 3-4 minutes on one side until browned. Flip the patties and place a slice of American cheese on each patty. Continue cooking for another 3-4 minutes, or until the desired doneness is reached.
Step 4: Assemble the Burgers. Lightly toast the burger buns, if desired. Spread a thin layer of butter on the inside of each bun. Place the cooked patties on the bottom half of each bun, top with the caramelized onions, and finish with the top bun. Serve hot and enjoy!