1Season the lamb neck liberally with salt and pepper. Heat the oil in a large heavy-bottomed pot or Dutch oven over medium high-heat. Sear the lamb until well browned on all sides. Remove to a platter and set aside.
2Reduce the heat to medium. Add the onion, carrot, and celery and sauté until softened but not browned, about 8 minutes. Add the garlic and continue to cook until aromatic, about 1 minute. Add the tomato paste and cook, stirring often, until the paste becomes a rich dark red, about 3 minutes.
3Deglaze the pot by adding the red wine, scraping up all the browned bits. Add the broth, tomatoes, thyme, rosemary, and bay leaf and stir to combine. Bring the mixture just to a boil and reduce the heat to low. Return the lamb and any drippings to the pot, nestling the pieces into the liquid until mostly covered. Cover securely and simmer for 2 to 3 hours, turning the lamb once per hour or until the lamb is tender and falling off the bone.
4Remove the lamb to a platter and let it cool slightly. Using two forks or your fingers shred the lamb from the bones into bite-sized pieces. Remove the thyme, rosemary, and bay leaf and discard.
5In a small saucepan bring the milk just to a simmer. Gradually add the milk to the sauce, a small amount at a time, stirring regularly. Return the lid to the pot, leaving slightly askew, and allow the mixture to simmer until the milk is well integrated, about 30 minutes. Return the lamb and any drippings to the pot, stir well to combine, and return just to a boil. (If the sauce seems too watery for your taste, lower the heat and simmer until adequately reduced.)
6Serve over freshly cooked pasta or polenta, garnished with fresh parsley.