Eggplant Parmigiana

Name: Eggplant Parmigiana
Ingredients: Extra-virgin olive oil; 2 large eggplant, about 2 pounds; Salt and pepper; 2 cups basic tomato sauce, recipe follows; 1 bunch fresh basil leaves, chiffonade; 1 pound fresh mozzarella, sliced 1/8-inch thick; 1/2 cup freshly grated Parmigiano-Reggiano; 1/4 cup fresh bread crumbs, lightly toasted under broiler; 1/4 cup extra-virgin olive oil; 1 Spanish onion, 1/4-inch dice; 4 garlic cloves, peeled and thinly sliced; 3 tablespoons chopped fresh thyme leaves, or 1 tablespoon dried; 1/2 medium carrot, finely grated; 2 (28-ounce) cans peeled whole tomatoes, crushed by hand and juices reserved; Salt