Combine beef with egg, Parmesan, parsley, garlic, salt and pepper in mixing bowl. Blend in breadcrumbs. Slowly add water, mixing until moist. Shape meatball mixture into 2 oz. patties. Refrigerate patties until ready to cook.
Grill meatball burgers over medium heat on lightly oiled grill. Cook thoroughly on each side. Add 1 slice of cheese, halved to fit, atop each burger. Cook 1 minute longer to melt cheese. Toast bread, cut side down on grill. Serve each cheeseburger with 2 Tbsp. grilled onions and sun-dried tomatoes.