Heat 1/3 cup of the olive oil in a 6-quart pot or Dutch oven, or other heavy-duty pot, over low heat. Add the onions and garlic, cover, and cook, stirring occasionally, until softened and sweet but not browned, about 15 to 20 minutes. Add the tomatoes, ¼ cup of the basil, ½ tsp. salt, and several grinds of black pepper. Simmer for 10 minutes to allow the flavors to develop.

Heat a grill on high heat. Place the vegetables on a large baking tray and toss with remaining 1/3 cup olive oil, ½ tsp. salt, and several grinds of black pepper. Oil the grill grate, and grill the pepper skin side down until the skin is charred and the flesh is softened, about 8 minutes. Grill the zucchini, squash, and eggplant skin side up until grill marks form, about 4 minutes. Flip the zucchini, squash, and eggplant and grill until grill marks appear and the vegetables are just softened, about 4 minutes more. Transfer back to the tray and allow to cool slightly.

Peel any charred skin off the pepper. Cut the vegetables into ½-inch chunks. Fold the vegetables into the sauce using a large spatula. Bring to a simmer, cover, and cook until the vegetables are tender, about 10 minutes. Fold in the remaining 2 Tbs. basil and season to taste with salt and pepper. Top with freshly grated Parmigiano-Reggiano and sprinkle with basil. Serve with crusty bread.