In a large enameled cast-iron casserole or Dutch oven, heat the oil until almost smoking. Season the lamb shanks with salt and pepper. Add them to the casserole and cook over moderately high heat, turning, until well browned all over.
Add the leek, onion, carrot, celery, garlic and thyme and stir for 2 minutes. Add the stock and bring to a simmer. Cover partially, reduce the heat to low and continue to simmer, turning occasionally, until the lamb is tender, about 2 hours. Remove the shanks to a plate.
Strain the cooking liquid and return it to the casserole along with the lamb shanks. Add the roasted tomato halves and garlic and simmer over low heat, basting the shanks occasionally, until the liquid is very flavorful, about 25 minutes. Ten minutes before the shanks are done, add the tarragon sprigs. Season with salt and pepper.
Discard the tarragon sprigs. Stir the arugula into the hot braising juices until wilted. Put 2 pieces of lamb shank, 2 tomato halves and 3 garlic cloves in each of 4 shallow bowls or soup plates. Spoon the Cannellini Beans into the bowls and ladle the arugula and juices over all.
The lamb shanks can be cooled and refrigerated for up to 3 days. Reheat gently before proceeding.
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