The secret to this hearty dish lies in a long, slow braising until the meat literally melts off the bone. Good wine and flavorful, homemade stock add a richness and complexity to the lamb that make for a truly memorable meal. The shanks can be braised either on the stovetop or in the oven. They can also be made a day or two in advance of serving and slowly reheated on the stovetop or in the oven. Serve with Creamy Polenta or with white beans that you've finished cooking in some of the braising liquid.
Combine the flour, salt, and pepper, and spread on a plate. Dredge the lamb shanks in the mixture, shaking off the excess flour.
Heat 2 tablespoons of oil in a large, heavy Dutch oven over medium-high heat. When hot, add the shanks and brown them on all sides, about 10 minutes. Remove from the pan with tongs and set aside.
Add the remaining 1 tablespoon of oil to the pan. Stir in the onions and rosemary, and sauté over medium heat until the onions start to soften and turn golden. Add the garlic and cook another 3 to 5 minutes.
Add the wine and balsamic vinegar, if using, and bring to a boil. Add the stocks and the tomato paste, and cook until the liquid is hot.
Add the shanks to the pan. If they're not completely submerged in the cooking liquid, add more stock or water to cover. Bring to a boil and then reduce heat to maintain a simmer. Alternately, transfer the Dutch oven to a preheated 350-degree F oven. Simmer, partially covered, until the meat is fork tender and easily slides off the bone, about 2 hours.
Remove the shanks from the braising liquid with tongs and place on a platter. Cover with foil to keep warm. Pass the liquid through a fine mesh sieve into a large pan, pressing hard on the solids to extract all the stock.
If you're not serving the shanks until the following day, refrigerate the shanks and the braising liquid separately, covered, overnight. Before reheating, skim off the layer of fat that rises to the top of the stock.
If you're continuing with the recipe, skim off as much fat as possible from the strained stock and discard. Bring the stock to a simmer over low heat and cook for 10 minutes to further concentrate its flavors. Return the lamb shanks to the pan and re-heat gently. Serve hot.