Material: Red Meat
Cooking: 30 Minutes
Preparation: 120 Minutes
Melt the butter in a large nonstick frying pan over medium heat until foaming. Add the onions and cook, stirring rarely, until deep golden brown and caramelized, about 40 minutes. Season generously 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 meat in a large bowl and add the measured 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 that is slightly wider in diameter than the slices of bread.
Heat a separate large frying pan (not the nonstick pan) over medium-high heat until a drop of water sizzles and immediately disappears on the surface. Place 2 of the 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 large plate and tent loosely with aluminum foil. Repeat with the remaining 2 patties; set aside.
On 1 side of each bread slice, spread about 1 heaping teaspoon of the mayonnaise in an even layer and all the way to the edges. Place 4 of the bread slices on a work surface mayonnaise-side down, and set the other 4 slices aside. Distribute half of the cheese among the 4 bread slices and divide the cooked onions evenly over the cheese, arranging the onions in an even layer. Top with the beef patties, divide the remaining cheese over the patties, and place the remaining 4 slices of bread on top mayonnaise-side up.
Heat the reserved nonstick pan over medium heat until a drop of water sizzles and immediately disappears on the surface, about 3 minutes. Place 2 of the 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 has melted, about 3 to 4 minutes more. Remove to a serving plate and repeat with the remaining 2 patty melts. Serve immediately.