Make small slits all over the meat with a sharp knife, inserting a sliver of garlic into each as you go. (If you can do this several hours or the night before cooking, all the better.) Rub the meat all over with salt and pepper. Heat the oven to 325ºF/160ºC.
Melt the butter in a deep, lidded casserole (cocotte), and brown the meat well on all sides. Add the onion, carrot, and herbs to the pot. Pour over the milk, and bring to a simmer on the stovetop. Cover the dish and transfer to the oven until tender, about 2 hours, turning the meat at least once.
Remove the meat from the pot and wrap in foil to keep warm. Remove the herbs, carrot, and onion and discard. The cooking juices will be curdled – ugly – but this is how they’re meant to be. Boil them down to about a cup/250 ml and purée with an immersion blender. Taste and adjust the seasonings.
Carve the meat and arrange in a serving dish. Pour over the sauce, sprinkle with the chives and/or parsley. Serve.