Preheat the oven to 325°. Season the shanks with salt and pepper. In a large enameled cast-iron casserole, heat the olive oil until shimmering. Add half of the shanks to the casserole and cook over moderately high heat, turning once, until browned, about 7 minutes. Transfer to a plate. Repeat with the remaining shanks.
Pour off all but 2 tablespoons of fat from the casserole. Add the onions, celery, carrots and a generous pinch of salt and cook over moderate heat, stirring, until the vegetables are very soft and golden, 15 minutes. Add the wine and bring to a boil. Simmer over moderate heat for 2 minutes. Add the tomatoes and broth and bring to a boil. Return the shanks and any accumulated juices to the casserole. Cover and braise in the oven for 3 hours, until the meat is very tender.
Using tongs or a slotted spoon, transfer the shanks to a plate and let cool slightly. Using 2 forks, shred the meat and scrape out marrow from the bones. Add the meat and marrow to the sauce and rewarm over moderately low heat. Season the sauce with salt and pepper and serve with polenta or pasta.