Patty Melt

Chop the onions into thin slices. Melt the butter in a large nonstick frying pan over medium heat until foaming. Add the onions and cook, stirring occasionally, until deep golden brown and caramelized, about 20 minutes. Season with salt and pepper, stir to combine, and transfer to a medium bowl; set aside. Wipe out the pan with paper towels and set aside.

While the onions are cooking, place the ground beef in a large bowl and add the remaining salt and pepper. Mix with your hands until just evenly combined. Divide into 4 equal portions and place on a baking sheet. Using your fingertips, press each portion into a 1/4-inch-thick patty slightly wider than the bread slices.

Heat a large frying pan over medium-high heat until a drop of water sizzles and disappears. Place 2 patties in the pan and cook undisturbed until browned on the bottom, about 2 to 3 minutes. Flip and cook until the patties are just slightly pink in the center for medium, about 2 to 3 minutes more. Transfer the patties to a plate and tent with foil. Repeat with the remaining patties; set aside.

Spread about 1 heaping teaspoon of mayonnaise on one side of each bread slice. Place 4 slices mayonnaise-side down on a work surface. Layer half of the cheese on the bread, divide the cooked onions evenly over the cheese, and top with the beef patties. Add the remaining cheese over the patties and top with the remaining bread slices, mayonnaise-side up.

Heat the reserved nonstick pan over medium heat until a drop of water sizzles and disappears, about 3 minutes. Place 2 sandwiches in the pan and cook until the bottom slices of cheese start to melt and the bread is golden brown, about 3 to 4 minutes. Flip and cook until the other side is golden brown and all of the cheese is melted, about 3 to 4 minutes more. Remove to a serving plate and repeat with the remaining sandwiches. Serve immediately.