Bring a 4-quart pot of water to a rolling boil, and cook the ziti al dente according to package directions.

While the water for the ziti is heating up, prepare the meat sauce. Coat a 4-quart pot with olive oil cooking spray, and place over medium heat. Add the ground beef, salt, pepper, and water. Cook, while stirring to crumble, until the meat is no longer pink. Transfer the cooked beef to a colander, and drain any excess fat.

Return the meat to the pot, and add the tomato sauce, carrot, garlic, onion, cinnamon, cloves, allspice, oregano, and water. Simmer over low heat, stirring occasionally, for about 10 minutes.

In a large saucepan, heat the milk until scalded (almost to boiling). Turn off the heat. In a small bowl, beat the egg and egg whites until light and frothy. Set aside. In another small bowl, dissolve the cornstarch in water, then add it to the milk, stirring constantly. Slowly stir the beaten eggs into the milk mixture, and add half of the cheese.

Preheat the oven to 350°F. Coat a 9-x-13-inch baking pan with olive oil cooking spray. Drain and rinse the cooked ziti.

Cover the bottom of the baking pan with half of the bread crumbs, then top with half of the ziti. Add the meat sauce in an even layer. Add the remaining ziti, then sprinkle with the remaining bread crumbs and top with the remaining cheese. Slowly pour the milk mixture over the casserole, making sure it is evenly distributed.

Cover the pan with foil, and bake about 1 hour, or until the top is light brown and firm.

Cool the casserole for 10 minutes before cutting into squares and serving.