How to Make It
Preheat the oven to 400°. Season the pork all over with salt and pepper. In a large skillet, heat the oil until shimmering. Add the pork roast and cook over moderately high heat, turning, until browned all over, about 12 minutes. Transfer the pork to a 12-by-14-inch roasting pan.
Pour off all of the fat from the skillet. Add the honey and cook over moderate heat, stirring, until it turns a deep amber, 3 to 5 minutes. Stir in the soy sauce, then add the demiglace and chicken stock and bring to a boil. Pour the liquid over the pork and roast for about 50 minutes, basting every 10 minutes, until an instant-read thermometer inserted in the thickest part of the meat registers 120°; add 1/4 cup of stock or water to the roasting pan if the juices evaporate too quickly.
Scatter the carrots, turnips and brussels sprouts around the pork and roast for about 25 minutes longer, basting occasionally, until the pork is glazed and an instant-read thermometer inserted in the thickest part registers 150°. Transfer the pork roast to a carving board, tent it with foil and let it rest.
Return the roasting pan to the oven and roast the vegetables for 20 to 25 minutes longer, until tender. Using a slotted spoon, transfer the vegetables to a medium bowl. Pour the pan juices into another smaller bowl or a gravy boat.
Thinly slice the pork and serve with the pan juices and vegetables.
Veal demiglace, a kind of concentrated stock, is available at Whole Foods and other specialty markets.