Place the zucchini and eggplant strips on a baking sheet, brush both sides with 1 tablespoon olive oil, and season with ¼ teaspoon salt and ¼ teaspoon ground pepper.
Place the onion wedges in a medium bowl, toss with 2 teaspoons olive oil and season with ¼ teaspoon salt and ¼ teaspoon ground pepper.
Grill the vegetables until tender, 2 to 3 minutes per side.
Sprinkle some flour on a flat surface, divide the dough into 6 pieces and roll into 6 rounds.
Before the pizzas go on the grill, be sure to have all of your ingredients ready and sitting right beside the barbecue.
Place the pizza dough rounds on the grill. Depending on the size of your grill, you may have to do this in two batches. Close the lid and grill for about 2 minutes, or until the dough is puffed and there are grill marks on the bottom of each round.
Flip the rounds and press down slightly with the back of a metal spatula to get rid of some of the air inside the dough. Immediately spread 1 tablespoon pesto on each pizza and top with the vegetables and cheese.
Close the lid and grill until the cheese melts, about 3 more minutes. Remove from the grill. Serve.