Trim the rib roast and tie it up with twine if you like. Season the roast liberally with salt and pepper. Use a small sharp knife to poke small holes all over the meat and insert the garlic slivers into the holes. Place the meat, bone side down, in a roasting pan or large cast iron pan. Toss the carrots, potatoes and onions with the olive oil, season with salt and pepper and surround the roast with them. Tuck the thyme sprigs amidst everything.
Place the roast in the oven and cook for 15 minutes (do not peek at the meat during this time; the high heat needs to be uninterrupted). Turn the heat down to 350°F and continue to roast for 30 minutes until a meat thermometer stuck into the very center of the roast and not near the bone registers 115°F (see Note). Turn the heat back up to 475°F and let it cook for another 10 minutes until it is beautifully browned on the outside. Check the internal temperature again: 125°F is rare, 130°F is medium rare, 135°F is medium rare. You do not want to go above 145°F—it’s kind of a waste of this cut of meat.
Remove the meat from the oven and transfer it to a cutting board; tent the meat with foil to keep it warm. Transfer the vegetables to a serving platter or bowl. Pour off all but one or two tablespoons of the fat from the roasting pan and place the roasting pan over a burner on high heat. Add the wine or broth and stir, scraping up any brown bits, until the liquid is reduced by half, about 7 minutes. Transfer it to a pitcher or bowl with a spoon to pass at the table.
When the roast has sat for 10 minutes to reabsorb the juices, slice it and moisten the meat a bit with some of the pan sauce before serving, then pass the rest of the sauce.