Trim the monk fish osso bucco of any excess membrane and dark tissue.
In a hot 12 in. sauté pan, sauté the onion, celery, carrots, and fennel until caramelized. Add garlic, ginger and add tomato paste and cook for 2-3 minutes. Deglaze with sauterne and bring to a boil. Add diced fresh tomatoes, chicken stock and lime juice. Reduce by half, and season with salt and pepper.
Season the monkfish osso bucco with salt, pepper. Sear cut side down over high heat in olive oil. Cook fish about 2 minutes on each side until caramelized.
Add the original sauce and cover. Place in a preheated 350ºF oven until the fish is cooked through, 10-15 minutes.
Remove from oven and remove the monk fish from the braising liquid. Keep warm.
Over medium heat, finish the sauce with butter. Check again for seasoning. Add the balsamic vinegar and more chicken stock if needed to adjust consistency. Place the monk fish in the center of the plate and pour the sauce over and around. Serve with farro, risotto, mashed potatoes, quinoa, rice or pasta.