Mix flour, yeast, rosemary, salt and sugar in a food processor. Pulse to evenly blend. Add warm water and oil and pulse until dough comes together in a ball.
Pulse dough with an on and off motion to knead, about 30 seconds. With lightly oiled hands, transfer dough to a lightly oiled bowl, turning to coat. Set aside to rest while preparing veggies.
Heat gas grill to medium-high or prepare charcoal grill with medium-hot coals. Place zucchini slices in a medium-size bowl. Secure onion slices with toothpicks, to keep rings from separating while grilling. Add to bowl with zucchini, and toss with balsamic vinaigrette.
Grill veggies, 2 minutes per side until tender and nicely marked. Transfer to cutting board; cool slightly, removing toothpicks from onions.
While vegetables cool, pat out dough to a 12 x 8-inch rectangle. Brush with 1 tablespoon oil, and transfer, oil side down, to grill.
Grill dough 3 minutes, or until dry on top and lightly browned on the bottom. Brush top with remaining tablespoon oil and flip over using a set of tongs and large spatula.
Top dough with sliced tomato and 1/4-cup cheese. Chop the vegetables, and sprinkle over dough. Top with remaining 1/4-cup shredded cheese and close lid on grill. Cook for 2 minutes, or until cheese is melted. Serve warm.