Ziti is a style of casserole, and this version is made with a rich, homemade tomato sauce, tiny Tuscan meatballs and ziti rigate pasta, topped with freshly grated Parmesan cheese.
Heat a medium Dutch oven over medium-high heat. Add 3 tablespoons of the olive oil and then the onion, garlic, carrot, celery and pepper flakes. Cook, stirring often, until the carrots are beginning to soften, about 2 minutes. Add 1/2 teaspoon salt and the tomato puree; rinse out each bottle of puree with 1/3 cup water and add that to the pot as well. Add the Parmesan rind and basil sprigs and stir to combine. Bring the sauce to a simmer, then reduce the heat to medium-low; simmer for 40 minutes.
Remove and discard the Parmesan rind, onion, celery, garlic and basil sprigs. Stir in 1 teaspoon salt and the remaining 3 tablespoons olive oil. Add the ricotta and whisk to combine. Set aside.
In a medium bowl, soak the bread in the milk and egg for 3 minutes, tearing apart the bread with your fingers. Add the Parmesan, parsley, 1/2 teaspoon salt, and ground beef. Mix gently with your hands just until combined.
Heat a large skillet over medium heat. Add enough oil to come 1 inch up the side of the pan. When the oil is hot, use a 1-tablespoon scoop to scoop and roll little meatballs. Working in several batches to avoid crowding the pan, add the meatballs to the hot oil and cook until evenly browned, about 3 minutes per side. Remove with a slotted spoon and transfer to a paper-towel-lined tray to drain.
Preheat the oven to 400ºF.
To assemble the dish: Cover the bottom of a 9-by-13-inch baking pan with 1 cup of the sauce. Next, layer half of the cooked ziti, followed by more sauce. Dot with three-quarters of the meatballs and all of the eggs, and sprinkle with half of the mozzarella and 1/2 cup of the Parmesan. Top with the remaining half of the pasta and then the remaining meatballs and mozzarella. Finish with the remaining sauce and 1/2 cup Parmesan.
Bake until lightly browned on top and bubbling around the edges, 30 minutes. Let sit for 10 minutes before serving.