Preheat the oven to 400 degrees. Put one rack in the middle of the oven and another in the upper third. Line two baking sheets with parchment paper, foil, or silicone baking mats for easy clean up.
Place the eggplant on one baking sheet, drizzle with 2 tablespoons olive oil and season with ½ teaspoon salt and ½ teaspoon pepper. Toss to coat. Add the rest of the vegetables to the other sheet pan, drizzle with 2 tablespoons of oil, ½ teaspoon salt, and ½ teaspoon pepper. Toss to coat.
Place the eggplant pan on the middle rack, the other pan on the upper rack and roast for 20 minutes, give the vegetables a toss and roast for another 15-20 minutes or until tender. (Can be done ahead of time, let cool, and store in a covered container in the refrigerator.)
Combine the roasted vegetables and pasta sauce in a large pot and bring to a simmer over medium-high heat. Lower the temperature and let simmer, stirring occasionally, for about 10 minutes. Season to tase with more salt and pepper, if needed. Stir in the basil just before serving.
While the sauce simmers, bring 8 cups of water and 2 teaspoons of salt to a boil. Slowly add the polenta while stirring. Cook, stirring, for 3 minutes. Stir in butter, if desired.
Serve the sauce over the polenta and top with shaved Parmesan cheese.