Make 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 sauté 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, 1 1/2 hours. Simmer mixture, uncovered, stirring occasionally, 1 hour more, or until lamb shanks are tender.
Make the gremolata while lamb is cooking: In a small bowl stir together gremolata ingredients.
Make 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 30 minutes. 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, and stir in butter and tarragon. Boil sauce, stirring occasionally, until thickened slightly. Strain sauce through sieve into a bowl and season with salt and freshly ground black pepper.
Sprinkle lamb shanks with remaining gremolata and serve with beans and sauce.