Preheat the grill or grill pan. Carefully oil the grill or grill pan with a cloth dipped in a little olive oil. If using a grill, have the grilling surface 8 to 10 inches above the embers. Regulate the amount of coals under the grill so that it maintains a consistent medium-hot heat (the oil on the grill will begin to smoke).
Prepare the chicken. Coat each chicken breast with the extra-virgin olive oil. Season each side evenly with a little salt and pepper. Place the chicken on the grill, skin side first, and cook for 4 to 5 minutes. Turn over and continue to cook in the same manner another 4 or 5 minutes until the meat is slightly firm to the touch. Season 1 side evenly with the roasted garlic rub. Remove from the heat and reserve until cool. Cut the chicken with the grain into bite-size pieces. Reserve.
Divide the dough into 6 balls. Working on a surface free of flour, roll each ball under your palm. As it rolls, it will stick slightly to the surface, creating tension that helps form a tight, round ball. Dust the work surface with flour, pat each ball down lightly, dust the tops with flour, cover with a towel, and let rise for about 15 minutes.
With a rolling pin, roll each ball into a circle 8 or 9-inches in diameter and about 1/8-inch thick and place on a cookie sheet that has been sprinkled with cornmeal (to keep the dough from sticking).
On a clean grill surface (use a wire brush then oil again as before) gently lay the piadine on the grill and cook until the bottom has set and is cooked, but not so well done that it is not pliable, about 4 minutes. Turn the crust and cook on the second side. The cooking time will vary with the heat of the grill. Remove crusts.
Meanwhile, toss together the spinach, mozzarella, tomato wedges, red onion, Parmesan and chicken. Toss with vinaigrette to coat the greens and taste for seasoning with salt and pepper. Place the crusts on plates and divide the salad among them. Serve "open face." Diners should fold their piadine in half and eat without utensils!