Bring a large pot of water to boil over medium heat. When boiling, salt the water and add the pasta. Cook until just shy of al dente.
Meanwhile, heat 2 tablespoons extra-virgin olive oil, 2 turns of the pan, in large skillet over medium heat. Add the onions, garlic and vegetables as you chop them, the zucchini, eggplant and red pepper. Season vegetables with salt and cover, cook until tender, about 10 to 12 minutes.
Just before draining the pasta, add about 1 cup starchy cooking liquid to the vegetables. Drain the pasta and add to the vegetables, tossing to combine. Stir in the basil and the cheese and a little black pepper, to taste. Transfer the pasta to a serving bowl and top with the shredded lettuce. Serve with an extra sprinkle of cheese and drizzle with the remaining 1 tablespoon of extra-virgin olive oil.