Trim and peel the eggplant and cut lengthwise into ¼-1/3-inch thick slices. Layer in a colander placed in the sink, sprinkling with a little salt in between the layers, and let stand 20 minutes to remove any bitterness. Meanwhile, preheat the grill to medium.
Pat the eggplant dry with paper towels and brush or drizzle each slice lightly with oil. Add to the hot grates and grill uncovered until marked underneath, 15 to 20 minutes. Turn the eggplant, cover and grill until nicely tender but not charred, about 5 minutes more. Transfer eggplant slices to a plate and let cool.
In a bowl, mix the ricotta and Parmesan and season to taste with salt and pepper. Stir in the egg and 2 tablespoons chopped fresh basil. Lay eggplant slices on a work surface (using 2 small pieces together if necessary) and top with a heaping tablespoon of the ricotta mixture. Cut the mozzarella crosswise into ¼-inch wide sticks, place 1 on top of each portion of cheese and roll up to enclose the filling. You should have 16 rolls; chop any remaining mozzarella for topping.
Preheat the oven to 375. Spread ¾ cup tomato sauce in the bottom of a 2½-quart shallow casserole. Arrange eggplant rolls in the baking dish in rows and spoon remaining 1¼ cups sauce on top.
Loosely cover rollatini with foil and bake for 25 minutes. Sprinkle any chopped mozzarella on top and bake uncovered until hot throughout and bubbly, 10 to 15 minutes more. Let stand 10 minutes. Sprinkle more chopped basil on top and serve.