How to Make It
Season the veal shanks with salt and pepper and dredge them in flour, shaking off the excess. In a large enameled cast-iron casserole, heat the olive oil until shimmering. Add the veal shanks and cook over high heat, turning, until browned all over, about 15 minutes. Transfer the shanks to a plate.
Add the garlic, carrots, celery, and onion to the casserole and cook over moderate heat, stirring frequently, until lightly browned, about 5 minutes. Add the wine and balsamic vinegar and cook over high heat, stirring occasionally, until the liquid has reduced to a thick syrup, about 25 minutes.
Return the veal shanks to the casserole and add the tomatoes and their liquid, the beef broth, 1 rosemary sprig, 2 tablespoons of the basil and the water. Season with salt and pepper and bring to a boil. Cover partially and cook over low heat, stirring occasionally, until the veal is very tender, about 2 1/2 hours.
Transfer the veal to a plate and let cool slightly. Discard the bones. Cut the meat into 1-inch pieces.
Strain the sauce through a fine sieve, pressing on the solids to extract as much liquid as possible; discard the solids. Rinse out the casserole and return the sauce to it. Boil the sauce over high heat until reduced to 2 1/2 cups, 8 to 10 minutes. Stir in the veal and the remaining 2 tablespoons of basil and season with salt and pepper. Keep the veal ragù warm over low heat.
In a large pot of boiling salted water, cook the pappardelle until al dente. Drain thoroughly and return to the pot. Add the veal ragù and toss gently. Transfer to a warmed large bowl, sprinkle with Parmesan and garnish with rosemary sprigs. Serve immediately, passing more Parmesan at the table.