Patty Melt

Melt 2 tablespoons of the butter in a medium-size cast-iron skillet over medium. Add onions, garlic, and 1/2 teaspoon of the salt. Cover and cook, stirring occasionally, until softened and translucent, about 5 minutes. Uncover and cook, stirring often, until golden brown and soft, 15 to 20 minutes, adding 1 tablespoon water at a time, as needed, if onions are looking dry. Remove skillet from heat, and stir in vinegar and Worcestershire. Transfer caramelized onions to a small bowl, and set aside. Wipe skillet clean.

While onions cook, divide ground beef into 4 portions. Press each into a flat, 1/4-inch-thick patty (just larger than the size and shape of the bread). Sprinkle evenly with pepper and remaining 1 teaspoon salt.

Heat oil in cleaned cast-iron skillet over medium-high. Cook 2 patties at a time, pressing gently, and flipping once, until browned on both sides but still pink in the center or to desired doneness, 3 to 5 minutes total. Transfer to a plate. Repeat cooking process with remaining patties.

Wipe skillet clean; reduce heat to medium-low. Spread 1 1/2 teaspoons of the mustard on each of 4 bread slices; top with 2 cheese halves, 1 patty, 1/4 cup caramelized onions, 2 more cheese halves, and remaining bread slices. Melt 1 tablespoon of the butter in skillet. Place 2 sandwiches in hot butter; cook (with another skillet on top of sandwiches to weigh them down) until golden brown and cheese is melted, about 2 minutes per side, adding 1 tablespoon butter to skillet when flipping to the other side and checking occasionally to prevent burning. Repeat cooking process with remaining 2 tablespoons butter and 2 sandwiches.