Prior to cooking, make sure that you have beef broth on hand, either purchased or homemade.
Removed the lamb shanks from the fridge, and allow to come up to room temperature, rinse and pat dry. You can leave them out while you start cooking the braising mixture.
In a large cast iron soup pot, melt the butter over medium heat.
Add the onion to the pot, and saute, adding a pinch of salt to season as well as allow the onions to release some liquid.
After a minute or so, add the garlic to the pot, and continue to saute.
After another minute, add the celery and rosemary to the pot, and continue to saute until the vegetables have softened.
Whisk the balsamic vinegar and coconut aminos together, and pour over the vegetables. Stir to evenly combine. Turn the heat down to medium-low, and let the vegetables cook in the liquid for about 3-5 minutes.
Add the beef broth to the pot, and stir to combine. Season with a pinch of salt and black pepper.
Add the tomato paste to the pot, and stir until it is evenly combined. Allow to cook for about a minute, and then remove from heat.
Season the lamb shanks with salt, and heat the tallow (or your choice of frying fat) in a cast iron skillet over high heat.
Sear the lamb shanks until brown on all sides, about 5 minutes total.
Transfer the lamb shanks to the pot with the braising liquid, turning to coat with liquid.
Bring lamb shanks to a boil, and then reduce to a simmer and cover.
Cook for about two and a half hours, checking every so often, and rotating the lamb shanks. They will be finished when the meat is tender and can fall off the bone.