1 cup crumbled ricotta salata or mild feta cheese (5 ounces)
Working over a strainer set over a bowl, seed the tomatoes; press on the seeds and pulp to extract as much of the juice as possible. Cut the tomatoes into 1/2-inch chunks and transfer them to the bowl with the tomato juice.
In a medium nonstick skillet, heat 3 tablespoons of the oil. Add the eggplant, season with salt and pepper and cook over moderately high heat, stirring occasionally, until tender and just beginning to brown, about 5 minutes. Transfer to a medium bowl.
Add the remaining 1 tablespoon of olive oil to the skillet, add the onion and cook over moderate heat until softened, about 4 minutes. Add all but 2 tablespoons of the tomatoes, 1/2 cup of water and salt and pepper. Cover and cook over low heat until the tomatoes are very soft, about 15 minutes. Transfer the sauce to a food mill and puree to remove the skins. Alternatively, press through a sieve with a rubber spatula. Return the sauce to the skillet, add the basil and all but 2 tablespoons of the eggplant and cook over low heat until warmed through.
Meanwhile, fill a large pot with water, cover and bring to a boil. Add salt and the pasta. Cover partially until the until the water just returns to a boil, then uncover, stir the pasta and cook until al dente. Drain well, reserving 1/2 cup of the cooking water, and transfer the pasta to a warmed large serving bowl. Add the sauce, three-quarters of the ricotta salata and a good grinding of pepper; toss well. Add some of the reserved cooking water if the pasta is dry. Garnish with the reserved tomatoes, eggplant and ricotta salata and serve immediately.