Preheat the oven: Adjust the oven rack to the lower-middle position and preheat to 450°F. Lightly oil a rimmed baking sheet with ¼ cup of olive oil.
Prepare the dough: Roll out the pizza dough into a 16-by-12-inch rectangle on a lightly floured counter. If the dough springs back, let it rest for 10 minutes before rolling again. Transfer the dough to the prepared sheet, stretching it to fit the corners. Brush the dough with 1 tablespoon of olive oil and cover with plastic wrap. Let it sit in a warm spot until slightly risen, about 20 minutes.
Make the sauce: In a food processor, blend the drained tomatoes, remaining 1 tablespoon of olive oil, vinegar, garlic, and oregano until smooth, about 30 seconds. Transfer the mixture to a liquid measuring cup and add reserved tomato juice until it measures 2 cups. Season with salt and pepper. (Sauce can be refrigerated for up to 1 week or frozen for up to 1 month.)
Prepare the vegetables: In a microwave-safe bowl, combine mushrooms, bell peppers, and onion. Cover and microwave, stirring occasionally, until softened and the mushrooms release their liquid, about 8 to 10 minutes. Drain the vegetables, pat them dry with paper towels, and let cool slightly. Add shredded mozzarella and toss to combine.
Par-bake the crust: Remove plastic wrap from the dough and, using your fingers, make indentations all over the surface. Sprinkle 1 cup of Parmesan cheese over the dough and bake until the cheese begins to melt, about 7 to 10 minutes.
Assemble the pizza: Remove the sheet from the oven and spread the tomato sauce evenly over the crust, leaving a 1-inch border. Bake again until the sauce is deep red and steaming, about 7 to 10 minutes.
Add toppings and final bake: Evenly distribute the vegetable-mozzarella mixture and the remaining ½ cup of Parmesan over the pizza. Bake until the cheese is golden brown and bubbly, about 12 minutes.
Finish and serve: Remove the pizza from the oven and let it rest for 5 minutes. Sprinkle with fresh basil, slice, and serve warm.