Preheat grill to medium-high. (For charcoal grilling or an oven variation, see below.)

Cut eggplant into 1/2-inch thick rounds. Grill, turning once, until marked and softened, 4 to 6 minutes. Let cool slightly, then thinly slice into strips. Reduce heat to low.

Sprinkle cornmeal onto a pizza peel or large baking sheet. Roll out the dough (see Tip) and transfer it to the prepared peel or baking sheet, making sure the underside of the dough is completely coated with cornmeal.

Slide the crust onto the grill rack; close the lid. Cook until lightly browned, 3 to 4 minutes.

Using a large spatula, flip the crust. Spread marinara sauce on the crust, leaving a 1-inch border. Quickly top with the eggplant, basil and garlic. Lay the Parmigiano-Reggiano shavings on top.

Close the lid again and grill until the cheese has melted and the bottom of the crust has browned, about 8 minutes. Variations: Pizza on a charcoal grill: Light 6 quarts (about 1 large chimney starter full) of charcoal and burn until the coals are mostly white, about 20 minutes. Spread the coals in an even layer. Place a grate over the coals. Let the coals burn until they are about medium-low. (Grill any toppings for the pizza while the coals are burning down.) To test the heat, hold your palm about 5 inches above the grill rack; if you can hold it there for about 8 seconds before you need to move it away, the fire is medium-low. Transfer the crust to the grill rack, cover the grill and cook the crust, checking once or twice, until lightly browned, 3 to 4 minutes. Flip the crust, quickly add the toppings, cover the grill and cook until the toppings are hot and the bottom of the crust has browned, 5 to 8 minutes. If your crust browns faster than your toppings are cooking, slide a baking sheet under the pizza to keep the crust from burning while the toppings finish. Pizza in the oven: Place a pizza stone on the lowest rack; preheat oven to 450 degrees F for at least 20 minutes. Roll out the dough and place on a cornmeal-dusted pizza peel or inverted baking sheet, using enough cornmeal so that the dough slides easily. Slide the dough onto the preheated stone and cook until the bottom begins to crisp, about 3 minutes. Remove the crust from the oven using a large spatula and place it uncooked-side down on the peel or baking sheet, making sure the underside of the crust is completely coated with cornmeal. Quickly add the toppings and slide the pizza back onto the stone. Continue baking until the toppings are hot and the bottom of the crust has browned, 12 to 15 minutes. Individual variation: The dough can be turned into 4 or 6 personal-size pizzas. After kneading, divide the dough into 4 or 6 equal balls. Brush with oil and place 3 inches apart on a baking sheet. Cover and set aside until doubled in size, about 1 hour. Roll each portion into a 6-to-8-inch circle.

Tips: Use a vegetable peeler to shave curls off a block of hard cheeses, like Parmigiano-Reggiano or Pecorino Romano cheese.

To roll out pizza dough: When you're ready to get your pizzas on the grill, turn the dough out onto a lightly floured surface. Dust the top with flour; dimple with your fingertips to shape into a thick, flattened circle-don't worry if it's not perfectly symmetrical. Then use a rolling pin to roll into a circle about 14 inches in diameter.