1) Place the meat on a baking tray lined with parchment or greaseproof paper and, using your hands, break it up into small pieces and combine evenly. Season the meat with salt and pepper to taste.
2) Divide the meat into 6 portions (about 115g 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.
3) Heat 1 tablespoon of the oil in a large frying pan over medium-heat. Add the onion, season with salt and pepper, and cook, stirring occasionally, until caramelised, about 7 minutes. Transfer the onion to a bowl.
4) Wipe out the frying pan; 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.
5) 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.
6) 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.