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.
(Or put sauce in the fridge and let the fat solidify overnight, remove fat, and
reduce. Keep meat separate)
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. Serve
sprinkled with Parmesan and garnish with rosemary sprigs.