Special equipment: a wood-fired pizza oven or a pizza stone; a pizza peel
Four hours before cooking, heat your wood-fired pizza oven. Alternatively, heat a pizza stone in a 500 degree F oven for 1 hour.
Add the eggplant to a colander and sprinkle with salt. Let it sit for 30 minutes to drain. Rinse off and pat dry.
Add the tomatoes to a blender and blend until smooth. Season with salt. Reserve.
Put the ricotta and goat cheese in a bowl and whip together with a rubber spatula or, if necessary, a hand mixer.
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.
Sprinkle some flour on a pizza peel. Using a rolling pin, stretch one of the dough balls to a very thin 10-inch round.
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.
Place the pizza on the pizza oven floor or on the pizza stone and bake until golden and crisp, about 3 minutes.