In a small bowl whisk the mayonnaise, basil, and garlic. Season with pepper. (The mayonnaise can be prepared up to 1 day in advance and stored, covered, in the refrigerator until ready to use.)
In a large nonstick skillet over medium-high heat, warm 2 tablespoons of the oil. Add the onions and sauté until they begin to brown, about 8 minutes, stirring frequently. Reduce the heat to medium-low and continue to sauté until the onions are very tender and caramelized, adding water by the tablespoonful as needed if the onions are dry, 15 to 20 minutes longer, stirring often. Add the balsamic vinegar, ½ teaspoon salt, and ¼ teaspoon pepper. Cook until the onions are coated with the vinegar and the mixture is moist, about 1 minute, stirring constantly. Remove from the heat and set aside.
Prepare the grill for direct cooking over high heat (450° to 550°F).
In a medium bowl gently mix the ground chuck with ¾ teaspoon salt and ½ teaspoon pepper. Shape the mixture into four patties, each about ¾ inch thick and about 3½ inches in diameter. Using your thumb or the back of a spoon, make a shallow indentation about 1 inch wide in the center of each patty. This will help the patties cook evenly and prevent them from puffing on the grill.
Brush the cut sides of the rolls with oil.
Brush the cooking grates clean. Grill the patties over direct high heat, with the lid closed as much as possible, until cooked to medium doneness, 8 to 10 minutes, turning once. During the last minute of grilling, place a slice of cheese on top of each patty to melt and toast the ciabatta halves, cut side down, over direct heat. Remove the patties and rolls from the grill.
Spread the basil mayonnaise on the cut side of each ciabatta half, dividing equally. Build each burger with a patty, prosciutto, onions, a tomato slice, and arugula. Serve right away.
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