How to Make It
Bring a large saucepan of water to a boil. Using a paring knife, core the tomatoes and mark an X on the bottom of each one. Add half of the tomatoes to the saucepan at a time and blanch until they begin to soften and the skins are wrinkled, 3 to 5 minutes. Transfer the hot tomatoes to a work surface and let cool slightly, then peel and coarsely chop. Working in 2 batches, puree the tomatoes in a food processor; you should have about 9 1/2 cups of puree.
In the saucepan, heat 2 tablespoons of oil. Add the onion and a generous pinch of salt and cook over moderate heat, stirring occasionally, until the onion is softened and beginning to brown, 8 minutes. Add the tomato puree and the basil sprig and bring to a boil. Simmer over moderately high heat, stirring frequently, until the tomato sauce is thickened and reduced to 5 1/2 cups, 5 minutes. Discard the basil and season the sauce with salt and pepper. Let cool slightly.
Meanwhile, in a medium saucepan, bring 1 1/2 cups of water to a boil. Add the rice and a generous pinch of salt and stir once. Cover and simmer over low heat until the rice is tender and the water is absorbed, about 20 minutes. Remove from the heat and let steam, covered, for 20 minutes. Fluff the rice with a fork and let cool.
Light a grill or preheat a grill pan. Brush the eggplant slices with oil and grill over moderately high heat, turning once, until tender and lightly charred, about 4 minutes per batch. Transfer the eggplant to a work surface and season with salt and pepper.
Preheat the oven to 350°. In a large bowl, combine the rice with 2 cups of the tomato sauce. Fold in the mozzarella, chopped basil and 1 cup of the Parmesan. Season the filling with salt and pepper.
Spread 3/4 cup of the tomato sauce in the bottom of each of two 9-by-13-inch baking dishes. Dollop 2 tablespoons of the rice filling near one end of each eggplant slice and roll into a tight cylinder; arrange in the prepared baking dishes. Spoon 1 cup of the tomato sauce over the involtini in each baking dish and top with the remaining 1/2 cup of Parmesan. Bake the involtini until the filling is hot and the sauce is just bubbling, 15 to 20 minutes. Let stand for 5 minutes before serving.