Bring 6 quarts of water to a boil in a large pot, and add 2 tablespoons salt.
Meanwhile, in a 10 to 12 inch sauté pan, heat 3 tablespoons of the olive oil over medium-high heat until almost smoking. Working in batches, sauté the eggplant slices, seasoning them with salt and pepper, turning once, until golden brown on both sides. Transfer to a plate lined with paper towels to drain.
Preheat the oven to 375 F. Grease a 9 by 12 inch baking dish with 1 tablespoon of the olive oil.
Cook the penne in the boiling water for 2 minutes short of the package instructions; it should still be quite firm. Drain and rinse under cold water until cool. Drain very well, place in a large bowl, and toss with 1 cup of the tomato sauce.
Cover the bottom of the baking dish with ¼ cup of the tomato sauce. Top with half the bread crumbs, then add half the pasta. Arrange half of the eggplant slices, overlapping them slightly, on top of the pasta. Dot about ¼ cup of tomato sauce over the eggplant, and top with half of the pecorino and half of the basil. Top with the remaining pasta, arrange the remaining eggplant over the pasta, and dot with the remaining tomato sauce. Sprinkle with the remaining pecorino and basil, and then the remaining bread crumbs, and drizzle with the remaining 2 tablespoons olive oil.
Bake for 45 minutes. Let rest for 10 minutes before serving. Place a generous portion of pasta on each plate, grate ricotta salata over, and serve.
In a 3 quart saucepan, heat the olive oil over medium heat. Add the onion and garlic and cook until soft and light golden brown, 8 to 10 minutes. Add the thyme and carrot and cook until the carrot is quite soft, about 5 minutes.
Add the tomatoes, with their juice, and bring to a boil, stirring often. Lower the heat and simmer until as thick as hot cereal, about 30 minutes. Season with salt. The sauce can be refrigerated for up to 1 week or frozen for 6 months.