Preheat the oven to 350 degrees F.
Lay the beef in a large roasting pan with the bone side down. (The ribs act as a natural roasting rack.) In a small bowl mash together the garlic, horseradish, salt, pepper, and olive oil to make a paste. Massage the paste generously over the entire roast. Scatter the vegetables and halved garlic around the meat and drizzle them with a 2-count of oil. Put the pan in the oven and roast the beef for about 1 1/2 to 2 hours for medium-rare (or approximately 20 minutes per pound). Check the internal temperature of the roast in several places with an instant-read thermometer; it should register 125 degrees F. for medium-rare. Remove the beef to a carving board and let it rest for 20 minutes. The internal temperature of the meat will continue to rise by about 10 degrees. Remove the vegetables and set aside. Pour the pan juices into a fat separator or small bowl and set aside to allow the fat and beef juices to separate. Pour off and discard the fat. You will use the tasty beef juices for the mushrooms.
Place a clean skillet over medium heat. Add the butter and a 2-count drizzle of oil. When the butter starts to foam. add the mushrooms and thyme; and season with salt and pepper. Stir everything together for a few minutes. Add the red wine, stirring to scrape up any stuck bits; then cook and stir to evaporate the alcohol. When the wine is almost all gone, add the reserved beef juices. Let the liquid cook down and then take it off the heat. Stir in the cream and chives, and season with salt and pepper.