Remove the rib roast from the refrigerator 30 minutes before you’re planning to cook it.
Preheat oven to 450°F.
In a small bowl, mix the softened butter, parsley, rosemary, thyme, salt, and black pepper.
Place the rib roast in a roasting pan. Spread the butter all over the rib roast. Roast for 30 minutes at 450°F.
Reduce the oven temperature to 350°F and cook until the roast reaches 120°F for medium rare (the temperature will rise to 130–135°F after it is removed from the oven). It will be approximately 15 minutes per pound of meat.
Turn the oven up to 400°F. Remove the roast from the oven and place on a cutting board. Tent with aluminum foil.
Transfer the beef drippings to a measuring cup (preferably one designed for gravy).
Place the carrots on a sheet pan and spoon 3 tablespoons of the beef fat (from the top of the cup) over the top. Toss to combine. Season with salt and roast for 20–25 minutes, or until tender.
Meanwhile, heat 1½ cups of beef drippings (from the bottom of the cup) in a saucepan over medium heat. If you don’t have enough drippings, add beef broth to reach 1½ cups.
Place the flour with ½ cup water in a jar with a tight fitting lid. Shake until the flour is incorporated. Once the beef drippings come to a boil, whisk in the flour mixture. Cook over medium heat, whisking occasionally, until the gravy is thick, about 5–6 minutes. Season with salt and pepper to taste.
Carve the prime rib into thin slices and serve with the roasted carrots and gravy.