In a small saucepan combine the seasoning ingredients. Whisk over low heat just until the oil becomes fragrant and the mixture is warm. Remove from the heat and let stand at room temperature for 2 hours.
Prepare the grill for direct cooking over medium heat (180° to 230°C).
Prepare the grill for cooking over medium heat (200°C).
Lightly flour one of the pizza dough balls and the back of a 38x25-centimeter baking sheet. Stretch out the dough to the size of the baking sheet and place it on the back of the floured sheet.
Whisk the seasoning mixture and then generously spread over the top side of the dough.
Brush the cooking grates clean. Turn the baking sheet over (like a big spatula) and place it directly on the cooking grate, dough side down. Remove the baking sheet and brush the top side of the dough with the seasoning mixture. Grill over direct medium heat, with the lid closed, until the underside of the dough is marked, 2 to 3 minutes. Don't worry if the crust bubbles; it will deflate when turned over. Slide the dough onto the baking sheet and flip the uncooked side onto the cooking grate. Brush the top with more of the seasoning mixture and continue to grill, with the lid closed, for 3 to 4 minutes.
Brush the cooking grates clean. Turn the baking sheet over (like a big spatula) and place it directly on the cooking grate, dough side down. Remove the baking sheet and brush the top side of the dough with the seasoning mixture. Grill over medium heat, with the lid closed, until the underside of the dough is marked, 2 to 3 minutes. Don't worry if the crust bubbles; it will deflate when turned over. Slide the dough onto the baking sheet and flip the uncooked side onto the cooking grate. Brush the top with more of the seasoning mixture and continue to grill, with the lid closed, for 3 to 4 minutes.
Transfer to a cooling rack. Top with chopped tomatoes, chopped onions, and fresh basil, if desired. Cut into individual pieces. Repeat the same grilling procedure for the two remaining balls of dough. Serve warm.
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