To make the lamb shanks: Pat lamb shanks dry and season with salt and pepper. In an 8 quart heavy flameproof casserole heat oil over moderately high heat until hot but not smoking and brown lamb shanks well in batches, transferring to a plate as browned. To casserole add onion, carrot, celery, and garlic and saute until onion is softened. Add wine and simmer mixture, stirring occasionally, until liquid is reduced to about 3 cups. Return lamb shanks to casserole and stir in broth, tomato paste, and thyme. Bring liquid to a boil and simmer, covered, stirring and turning lamb shanks occasionally, 2 1/2 hours. Simmer mixture, uncovered, stirring occasionally, 1 hour more, or until lamb shanks are tender.

To make the Gremolata while lamb is cooking: In a small bowl stir together gremolata ingredients.

To make the Beans while lamb is cooking: In a saucepan heat oil over moderately high heat until hot but not smoking and cook onions, carrots, celery, and garlic, stirring, 2 or 3 minutes, or until softened. Add beans, 2 cups broth, butter, and bay leaf and cook over moderate heat, stirring occasionally and adding enough remaining broth to keep beans moist and to reach a creamy consistency, about 60 minutes (make sure they are tender). Discard bay leaf and add half of gremolata and salt and pepper, to taste.

Transfer lamb shanks to a plate and keep warm, covered with foil. Strain braising liquid through a sieve into a saucepan, discarding solids. Boil sauce, stirring occasionally, until thickened slightly. Pour sauce into a bowl and season with salt and freshly ground black pepper. Stir in the butter and tarragon.

Sprinkle lamb shanks with remaining gremolata and serve with beans and sauce.