Four hours before cooking, heat your wood-fired pizza oven. Alternatively, heat a pizza stone in a 500 degree F oven for 1 hour.
Heat a grill or grill pan over medium-high heat. Grill the onion, zucchini and eggplant until charred and cooked through, about 2 minutes per side; the eggplant will need a little longer. Drizzle with olive oil and season with salt and pepper.
Ladle about one-quarter of the sauce on the dough and spread around, leaving an inch of crust around the edge uncovered. Sprinkle one-quarter of the cheese mixture over the sauce and lay out one-quarter of the vegetables on top of the cheese.