Preheat the oven to 400 degrees F.
Toss the potato, yam, bell pepper, carrots, and 2 tablespoons of olive oil in a 13 by 9 by 2-inch baking dish to coat. Sprinkle with salt and pepper and toss until coated. Spread vegetables evenly over the bottom of the pan.
Arrange the onion slices evenly over the vegetable mixture. Arrange the zucchini over the onion. Drizzle with 2 tablespoons of oil. Sprinkle with salt and pepper. Arrange the tomato slices over the zucchini.
Stir the Parmesan and bread crumbs in a small bowl to blend. Sprinkle the Parmesan bread crumbs over the vegetables in the baking dish. Drizzle with the last tablespoon of olive oil.
Bake uncovered until the vegetables are tender, and the topping is golden brown, about 40 minutes. Garnish with fresh basil sprigs, if desired.