Place the meat on a parchment paper-lined baking sheet and, using your hands, break it up into small pieces and combine evenly. Season the meat with salt and pepper to taste.
Divide the meat into 6 portions (about 4 ounces each). Using your hands, form each portion into a ball shape by gently tossing it from one hand to the other. Form each portion into a thin patty. Each patty should be somewhat larger than a slice of the rye bread since it will shrink when cooked.
Heat 1 tablespoon of the oil in a large cast-iron skillet over medium-heat. Add the onion, season with salt and pepper, and cook, stirring occasionally, until caramelized, about 7 minutes. Transfer the onion to a bowl.
Wipe out the skillet; return to the heat, and heat 1 tablespoon oil. Add 2 patties and saute, turning once, until well browned and cooked to medium, about 1 minute per side. Transfer to a plate and repeat with the remaining oil and patties.
Arrange 1/6 of the onion over one bread slice. Place 1 cheese slice over the onion and top with a patty. Top the patty with a slice of cheese and cover with a bread slice. Spread about 2 teaspoons of butter over the outside of each slice. Repeat with the remaining ingredients to make 6 sandwiches.
Heat the cast-iron skillet over medium heat. Add 2 sandwiches and cook, turning once, until lightly browned and the cheese has melted, about 5 minutes per side. Transfer to a cutting board and repeat with the remaining 4 sandwiches. Cut sandwiches in half crosswise and serve.
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