Special equipment: 6 (5-by 2-inch) ceramic souffle dishes
Preheat oven to 350 degrees F.
Place the eggplant slices on top of a sheet tray lined with parchment or waxed paper. Drizzle with oil and season with salt and pepper. Bake in oven until it turns golden brown, about 20 minutes. Remove from oven and let it cool.
Oil the bottoms and sides of 6 souffle dishes and layer 4 to 5 pieces of eggplant in a circular pattern into each souffle dish, letting excess drape out over the edge.
Heat up the tomato sauce in a large saucepan. Then turn off flame. Reserve some tomato sauce for plating in a smaller pot.
In a large pot with boiling salted water, cook the cavatelli pasta for 3 to 4 minutes, and drain.
In the pot with the sauce, add the partially cooked cavatelli, mozzarella, and pecorino to the tomato sauce and cook over medium heat just to blend well. When hot, stir in the basil chiffonade. Spoon the pasta-sauce mixture into the souffle dishes and fold over the top with excess eggplant. Bake in oven for 4 to 5 minutes.
To serve, spoon the remaining tomato sauce onto individual plates. Turn a finished timballo upside down onto a large spatula to remove it from the souffle dish, then carefully slide it onto the plates. Sprinkle with pecorino and garnish with a basil leaf.
In a heavy 5-quart saucepan cook onion, carrot, celery, garlic, basil, bay leaf, and salt and pepper, to taste, in oil over moderate heat, stirring, until vegetables are softened. Stir in tomato paste and tomatoes with reserved juice.
Simmer sauce, covered, over moderately low heat, stirring occasionally, 35 minutes and simmer, uncovered, stirring occasionally, 15 to 20 minutes, or until thickened. Discard bay leaf. Sauce may be made 2 days in advance and kept covered and chilled.