When we were first married, living as graduate students in Ithaca, New York, we once decided to splurge and buy a prime rib of beef. It cost half a months' salary! I was so nervous about the preparation that I sat for nearly the whole roasting time, on the floor, in front of the oven, watching the roast. It was delicious. Over the years I've perfected the recipe and become more confident. At the same time, fortunately, our status went from graduate students to actually having permanent jobs - so it's not quite as traumatic to buy a roast. I highly recommend making this rare or medium rare. Prime rib is not meant to be cooked to well-done. Definitely use a meat thermometer and start with the roast near room temperature. Serve with an excellent bread and a simple vegetable, followed by a crisp salad. Pamper yourself with a chocolate dessert.
Roast the beef for 15 minutes. Reduce the oven temperature to 325°F. Continue to roast until the meat thermometer registers about 120°F. for medium-rare. (The meat will continue to cook for a few minutes outside of the oven.) Remove to a carving board and tent the roast loosely with foil to keep warm. A 9-pound roast will take about 2 hours and 45 minutes to reach medium rare, but you need to keep a careful watch as this time varies with the original temperature of the meat and with the exact fat quantity and shape of the roast.
When the beef is nearly done, begin to make the sauce: remove about 1 T. of the fat from the roasting pan, and put it into a small sauce pan. Heat the sauce pan over medium-high heat. Add the shallots and saute until tender. Turn off the heat and wait until the roast is finished in the oven.
As soon as the beef is tented, Pour any juices from the roasting pan into a glass measuring cup. Put this in the freezer while you are proceeding with the sauce. Add about 1/2 cup of broth to the roasting pan, heat the roasting pan over medium heat, scraping any browned bits from the bottom of the pan. Remove the roasting pan from the heat, and pour its contents into the sauce pan. Also add the beef broth and cognac or brandy to the sauce pan. Boil this mixture until the liquid is reduced to 2 cups. This will take about 15 minutes. Remove the pan juices from the freezer, discard any accumulated fat, and add the juices to the saucepan. Bring to a boil, and transfer to a sauceboat. Slice the roast at the table and serve with the prepared sauce.