Position the oven racks on the lowest and uppermost rungs of the oven. Place 2 large baking sheets in the oven and preheat to 500°F.

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.

Combine 1-1/2 cups of the roasted peppers, the garlic paste, and salt and pepper to taste in a blender or food processor and puree until smooth. Cut the remaining 3/4 cup peppers into long, narrow strips. Set aside.

With a rolling pin, roll each ball into a circle 8 or 9 inches in diameter and about 1/8 inch thick. Spread each round with about 3 tablespoons of the red pepper puree, then sprinkle with 1 tablespoon Parmesan and 1-1/2 teaspoons oregano.

Remove the baking sheets from the oven, sprinkle evenly with cornmeal, and transfer the dough rounds to the sheets. Bake until slightly underdone (they will be lightly browned around the edges but still pliable), 8 to 12 minutes.

While the crusts are baking, in a bowl, toss together the spinach, vinaigrette, mozzarella, reserved pepper strips, and chicken. Taste for seasoning. Let the crusts cool very briefly so they won't cook the greens when filled. Transfer the crusts to plates and divide the salad among them. Serve "open face." Diners fold their piadine in half.