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1. Prepare a charcoal or gas grill for direct grilling over high heat. Brush and oil the grill grate.
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2. In a large bowl, using a spoon, not your hands, gently mix together the chuck and sirloin. Run your hands under cold water and divide the mixture into 6 equal portions. Shape each portion into a patty about the size and shape of a slice of bread, being careful not to compact the meat too much. Season on both sides with salt and pepper. Make a depression in the center of each patty with your thumb. Refrigerate the patties until the grill is ready.
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3. Brush both sides of each bread slice with the butter.
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4. Place the patties, indent side up, and onion slices on the grill directly over the fire. Cook, turning once, until both the patties and the onions are nicely charred on both sides and the burgers are cooked to your liking, 4-5 minutes per side for medium. During the last 2 minutes of cooking, top each patty with a slice of cheese and put the bread slices along the edge of the grill, then cover the grill. Grill, turning the bread slices once, until the cheese is melted and the bread is lightly toasted.
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5. Place each burger on a slice of toasted rye. Toss the onion slices to break up the rings, and divide evenly among the burgers. Top each burger with a second slice of toasted bread and serve at once.
