Prepare the Sauce: Adjust oven rack to upper-middle position and preheat oven to 500°F. In a food processor, blend tomatoes, 1 tablespoon oil, vinegar, oregano, and garlic until smooth (about 30 seconds). Transfer to a measuring cup and stir in reserved tomato juice until sauce measures 1 cup. Season with salt and pepper.
Sauté Vegetables: Heat 1 tablespoon oil in the skillet over medium heat. Add mushrooms, bell pepper, and onion, cooking until softened (6–8 minutes). Drain vegetables, pat dry with paper towels, and transfer to a bowl. Let cool slightly, then mix with mozzarella. Wipe skillet clean.
Prepare the Dough: Grease the skillet with 2 tablespoons oil. Lightly coat the dough ball with flour and flatten it into an 8-inch disk with your fingers. Roll into an 11-inch circle, dusting with flour as needed. If dough shrinks back, let it rest for 10 minutes before rolling again.
Assemble the Pizza: Transfer dough to the prepared skillet, reshaping as needed. Spread ½ cup of sauce in a thin layer over the dough, leaving a ½-inch border. Evenly distribute the vegetable-mozzarella mixture and sprinkle Parmesan over the top.
Cook on Stovetop: Set skillet over medium-high heat and cook until the edges set and the bottom crust is spotty brown (about 2–3 minutes).
Bake the Pizza: Transfer the skillet to the oven and bake until the crust is golden brown, about 7–10 minutes.
Serve: Using potholders, remove the skillet from the oven and slide the pizza onto a cutting board. Let cool slightly, sprinkle with basil, slice, and serve.