To prepare dressing, combine first 8 ingredients in a large bowl, stirring with a whisk. Set aside.
Preheat grill to medium-high heat.
To prepare flatbreads, turn dough out onto a lightly floured surface; let stand for 15 minutes. Pat each dough portion into a 5-inch circle. Brush with 1 tablespoon oil. Place dough on grill rack coated with cooking spray; grill 2 minutes on each side or until crisp and well marked. Remove from grill; rub top of each flatbread with cut sides of 1 garlic clove. Tear each flatbread in half.
To prepare salad, lightly coat eggplant and the next 3 ingredients (through onion) with cooking spray. Arrange eggplant and zucchini on grill rack coated with cooking spray; grill 1 1/2 minutes on each side or until tender and well marked. Arrange bell pepper and onion on grill rack; grill 3 minutes on each side or until well marked.
Separate onion slices into rings. Cut the bell pepper into thick strips. Add onion, eggplant, zucchini, bell pepper, green olives, and tomato to dressing; toss gently to combine. Arrange the vegetables on a platter. Top with nuts, basil, cheese, and caper berries. Serve with flatbreads.