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1
Prepare a medium-high gas or charcoal grill fire. Alternatively, position an oven rack 5 to 6 inches from the broiler and heat the broiler to high. Line the bottom of a broiler pan with foil and lightly oil the perforated part of the pan.
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2
Meanwhile, heat the oil in a 10-inch skillet over medium-high heat. Add the onion, 1/4 tsp. salt, and 1/8 tsp. pepper; reduce the heat to medium low and cook, stirring occasionally, until deeply golden brown and tender, 15 to 18 minutes.
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3
Combine the mayonnaise, Dijon, lemon juice, rosemary, and garlic in a small bowl. Season to taste with salt and pepper and set aside.
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4
In a medium bowl, gently combine the beef with 1/4 tsp. salt and 1/8 tsp. pepper. Form the beef into 4 patties (3-1/2 inches in diameter) and make a deep depression in the center of each patty so the burgers keep their shape during cooking. Lightly sprinkle the patties with 3/4 tsp. salt and 1/2 tsp. pepper. Grill or broil them on the prepared pan for about 4 minutes per side for medium, or until desired doneness. Top each burger with 1 slice of the cheese and grill or broil until melted, 30 to 60 seconds.
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5
Toast the buns on the grill or under the broiler until golden, 30 to 60 seconds. Serve the burgers on the toasted buns with the caramelized onions, mayonnaise, and arugula.
