If you're using a baking steel or stone, place it on a rack in the bottom third of the oven. Set the other rack in the center of the oven and preheat the oven to 400°F.
Trim and chop the eggplant, pepper, zucchini, onion, and tomato into small dice, all around the same size. Spread out the chopped vegetables in a single layer on a large roasting pan or baking sheet. Scatter the oregano over top, season to taste with salt and pepper, and drizzle with the olive oil. Toss the vegetables gently to coat in the oil, then spread them out again in a single layer (or as close to a single layer as you can get).
Roast the vegetables for 40 minutes or until they have cooked down and are golden brown in spots. Remove the pan from the oven and set it aside to cool while you roll out the pizza dough and grate the cheeses. (Or roast the veggies ahead of time and store them in the fridge until you're ready to make the pizza. They will last in the fridge for up to 1 week.)
If you're using a baking sheet for your pizza, increase the oven temperature to 500°F. If you're using a baking stone or steel, increase the oven temperature to 550°F. Let the oven heat up for about 20 minutes. Then, if you’re using a baking stone or steel, switch the oven to Broil on high.
Stretch or roll out your dough to a 12-inch circle, then transfer it to an oiled or parchment-lined baking sheet or a lightly floured pizza peel (if using a baking stone/steel).
Scatter the mozzarella cheese evenly over the dough, then scatter the roasted vegetables over top of that. Sprinkle with the Parmesan cheese.
Transfer the pizza to the oven and bake until the crust is golden and the toppings have browned in spots—8 to 10 minutes on the baking sheet, 6 to 8 minutes on the baking stone/steel. Remove the pizza from the oven.
Drizzle the pizza with a little olive oil and season with a pinch of salt and a grind or two of black pepper. Slice and serve.