In an 8 qt. container, add red wine, vinegar, onion, carrot, celery, and pickling spice; stir to combine. Add lamb shanks, cover with lid, and marinate for 3 to 5 days, turning each day.
Preheat oven to 325°F.
Remove lamb from marinade, transfer to a sheet pan lined with parchment paper, and pat dry.
In a medium braising pan over medium-high heat, add oil and sear lamb shanks on all sides until golden brown, about 8-10 minutes. Remove lamb shanks and transfer to a sheet pan. Add remaining onion, carrot, celery, and garlic to the pan and cook until vegetables are caramelized, about 4 minutes.
Strain marinade, add to braising pan, and reduce by half. Add lamb shanks back to the pan and cover by three-quarters with the brown stock. Cover pan with aluminum foil and place in the oven. Bake for 2 hours or until lamb is tender, but not falling off the bone.
Remove lamb from the braising liquid and set aside. Strain liquid and reduce to preferred flavor profile. Thicken if necessary, using a cornstarch slurry. Return shanks to liquid and finish with parsley, tarragon, and butter.