Heat the oven to 325°F. Season beef with 1/2 teaspoon salt and pepper and brown in oil in a Dutch oven for 6 to 8 minutes, turning. Transfer to a plate.
Cook carrot and onion in the pot for 5 minutes, then stir in garlic and tomato paste for 1 minute. Return beef and add tomatoes, broth, water, thyme and remaining salt.
Cover and braise for 2 1/2 to 3 hours until the beef shreds easily. Shred it into short strands in the sauce, discarding large pieces of fat. Increase the oven to 375°F.
Boil rigatoni in water with the tablespoon salt until 3 minutes short of the package time. Reserve 1 cup pasta water, drain and fold into the ragù with 1/2 cup reserved water; use more only if the sauce is too thick to spread.
Transfer to a deep 9-by-13-inch baking dish and scatter with mozzarella and Parmesan. Cover with foil and bake for 25 minutes.
Uncover for 8 to 10 minutes until the center reaches 165°F and the cheese browns lightly. Rest for 10 minutes before serving.