Prepare the sauce according to the directions. You can make this ahead and store in the refrigerator or freezer. If frozen, thaw before using.
Preheat the grill to medium, about 350° F/175° C.
Slice the eggplant into 1/4-inch (0.5 cm) rounds with the skin still on. Brush with olive oil. Grill for about 3 to 4 minutes per side.
Preheat the oven to 350° F/175° C.
In the medium mixing bowl, stir together the ricotta, egg, grated Parmesan, and dried basil. If using fresh basil, leave it out for now.
Spread a ladle or two of the prepared sauce in the bottom of the baking dish. Lay down a row of the grilled eggplant, then spread about 1/3 of the ricotta cheese mixture. It doesn’t have to look pretty, just make sure that it is evenly distributed, even if it’s in clumps. If using fresh basil, add 1/2 of the leaves in a layer.
Add about 2 more cups of sauce, then half the remaining grilled eggplant, 1/2 of the remaining cheese mixture, and the rest of the fresh basil (if using).
Add another 2 cups of the sauce, the rest of the grilled eggplant, and the rest of the ricotta cheese mixture. Spread the remaining sauce on top then top with the shredded mozzarella and some more Parmesan.
Bake uncovered in the preheated oven for 30 to 35 minutes, until the lasagna is cooked all the way through and the top is bubbly and golden brown. Remove from the oven and allow to sit for about 5 to 10 minutes before serving.