In a large Dutch oven or other heavy, lidded pot, heat the olive oil over medium-high heat. Salt the shanks, then brown them well on all sides except the one with the bone; this helps the shanks hold together better. Preheat the oven to 325°F.
When the shanks are nicely browned, remove them to a bowl. Add the carrots, onion and celery and sauté until slightly browned, about 6 to 8 minutes. Add the garlic cloves and cook another minute, stirring often. If you are cooking lamb shanks, leave the vegetables in. If you are cooking deer or other wild game shanks, remove the vegetables.
Add the tomato paste, wine, thyme and oregano to the pot. Mix well, and once it thickens, add a cup of water and mix well. Add the venison shanks back in now, along with any juice that has accumulated in the bowl. Cover with more water by just a little, bring to a simmer and add salt to taste. Cover the pot and let this cook in the oven for 90 minutes.
At this point, lamb shanks will be done. Deer shanks will still need some time, but you can add back the vegetables now. Keep cooking until the meat wants to fall off the bone, anywhere from another 30 minutes to another 2 hours. It's wild game, so you never know...
When the meat is tender, remove it gently to an ovenproof pan, along with all the vegetables. Turn off the oven and set this pan in the oven to keep warm. Set the pot on a strong burner and boil down the braising liquid. This will take roughly 20 to 30 minutes, so now is a good moment to make rice or mashed potatoes.
When the sauce has reduced and thickened a bit, turn off the heat. When it stops, bubbling, add the cold butter a tablespoon at a time, swirling until it disappears. Use this as your sauce for the shanks and vegetables. Top it all with fresh herbs and black pepper.