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Cooking a patty melt in a cast-iron pan gives the sandwiches crispy, savory bits of meat and cheese around the edges. Caramelized onions deliver tangy-sweet flavor.
Gather all ingredients.
Heat olive oil in a large cast iron skillet over medium. Add onions and 1 teaspoon salt; cook 20 minutes, stirring often, until well browned. Transfer onions to a medium bowl and stir in Worcestershire sauce. Do not wipe pan clean.
While onions cook, divide ground beef into 4 portions. Press each into a thin patty the same size as the bread between two pieces of parchment paper. Sprinkle evenly with remaining 1 teaspoon salt.
Cook patties two at a time, pressing slightly, until just barely pink in the middle, about 2 to 3 minutes per side. Remove from pan and wipe pan clean.
Assemble sandwiches. Spread a thin layer of mayonnaise on the outsides of each piece of bread. Divide American cheese slices among 4 slices rye bread, spacing evenly on the bread slices. Top each with a burger patty, 1/4 of the cooked onions, and Swiss cheese slices. Finish with remaining bread slices.
Melt 2 tablespoons butter in pan. Add 2 sandwiches and cook over medium until browned on each side and cheese has melted, about 3 to 4 minutes per side. Repeat with remaining sandwiches and serve immediately.