Preheat the oven to 425 degrees F. Lightly grease a large sheet pan.
Place the halved red potatoes on the pan. Toss them with 1 tablespoon olive oil, 1/4 teaspoon salt, 1/4 teaspoon black pepper, and 1/2 teaspoon garlic powder. Turn the potatoes cut side down.
Roast the potatoes for 15 minutes.
While the potatoes roast, toss the chicken with 1 tablespoon olive oil, the Italian seasoning, paprika, onion powder, remaining garlic powder, remaining salt, and remaining black pepper.
Add the chicken to the pan with the potatoes. Spread everything into a single layer and roast for 10 minutes.
Toss the eggplant, zucchini, and yellow squash with the last 1 tablespoon olive oil, plus a pinch of salt and black pepper. Add them to the pan and roast for 10 minutes.
Scatter the cherry tomatoes over the pan. Roast for 8 to 12 minutes more, until the chicken reaches 165 degrees F and the potatoes and squash are tender.
Sprinkle with grated Parmesan cheese before serving, if you like.