Heat the oven to 425°F and line a rimmed baking sheet with parchment. Toss the tomatoes, zucchini, and corn with 1 tablespoon olive oil, the minced garlic, about 1/4 teaspoon kosher salt, and the black pepper.

Spread the seasoned vegetables in a single layer on the prepared pan. Roast for 20–25 minutes, stirring once, until the tomatoes soften and begin to blister and the zucchini is lightly browned; drain off excess liquid if needed.

Reduce the oven temperature to 400°F. Place the two flatbreads on a baking sheet and brush each lightly with the remaining 1 tablespoon olive oil.

Sprinkle the mozzarella and Parmesan evenly over the flatbreads. Distribute the roasted vegetables over the cheese and add the goat cheese or feta if using.

Bake for 8–12 minutes, until the flatbread edges are crisp and the cheese is melted and lightly browned. Finish with fresh basil and red-pepper flakes if desired, then slice and serve hot.