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.

Meanwhile, heat the oil in a 10-inch skillet over medium-high heat. Add the onion, 1/4 teaspoon salt, and 1/8 teaspoon pepper; reduce the heat to medium low and cook, stirring occasionally, until deeply golden brown and tender, 15 to 18 minutes.

Combine the mayonnaise, Dijon, lemon juice, rosemary, and garlic in a small bowl. Season to taste with salt and pepper and set aside.

In a medium bowl, gently combine the beef with 1/4 teaspoon salt and 1/8 teaspoon 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 teaspoon salt and 1/2 teaspoon 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.

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.