Sprinkle the slices of eggplant with sea salt, then lay them in a wide colander. Cover tightly with a plate and place a weight on top of the plate. Stand the colander in the sink or over a basin for 1 hour to let the bitter juices of the eggplants drain away.
Meanwhile, make the tomato sauce. Place the onion, carrot, celery, garlic, and tomatoes in a heavy saucepan. Cover with a lid and place the vegetables over a low heat to cook in their own juices gently for about 40 minutes, or until all the vegetables are soft.
Push the softened vegetables though a food mill, or process in a food processor until puréed. Return the sauce to the heat to boil again until reduced to a thick texture.
Remove the saucepan from the heat, add salt and pepper to taste and reserve until required.
Rinse the eggplant slices and pat dry, then cut them into neat cubes.
Heat the vegetable oil in a large skillet until sizzling hot. Add the eggplant cubes and fry until soft and shiny. Drain the eggplant thoroughly on paper towels and keep warm.
Bring a large saucepan of salted water to a boil. Add the pasta, stir and return to a boil, and cook until al dente.
Reheat the tomato sauce gently and stir in the olive oil, then remove from the heat.
Drain the pasta and return it to the hot pan. Now you are ready to quickly assemble the dish.
Add alittle amount of the tomato sauce to the pasta, just enough to coat it lightly. Arrange the pasta in a high mound on a warmed platter. Pour the remaining sauce over the top so that it trickles down. Arrange the hot fried eggplant cubes down the sides of the mound. Arrange the basil sprigs half buried in the tomato sauce, then sprinkle the top with the cheese and serve immediately.