Take the roast out of the fridge 2 hours before you start cooking to bring the meat up to room temperature. This will aid in even cooking.
Heat the oven to 450° F. Season the rib roast with salt and pepper on all sides.
Using a mini processor or blender, blend the garlic, parsley, basil, and olive oil into a paste. Mix in the dijon and whole grain mustard.
Place the roast fat side up on a cutting board and rub the mix on the top and the sides. Set aside.
Cut the carrots into pieces about the size of your pinky finger, or about 2 inches long and about 1/4 inch thick.
Put all the vegetables - celery, carrots, onions, mushrooms, potatoes, and garlic - in a deep roasting pan that will fit the roast. Season with salt and pepper and mix well.
Pour in the red wine and chicken stock and toss in the bay leaves. Tie the thyme and parsley together and place on top of the vegetables in the middle of the pan.
Next, place the roast on top of the vegetables in the middle of the pan with the rub facing up.
Place the roasting pan in the oven set to 450° for 15 minutes. This will give the roast a nice brown color. Remove the roast from the oven and baste the meat with the juices from the roasting pan, about 2-3 spoonsful with a serving spoon.
Reduce the heat to 350° and return the roast to the oven. Your overall cook time will be about 12 minutes per pound for medium-rare or 15 minutes per pound for medium. See the "Recipe Notes" below.
Test the temperature with a meat thermometer. You want 115-120° for medium-rare or 130-135° for medium.
Take the roast out of the oven once the temperature has reached your desired doneness. Let the roast rest for 20 minutes before carving it. The internal temperature will continue to rise out of the oven.
Remove the roast to a cutting board and pour the juices into a sauce pan. Taste the vegetable to see if they need any additional salt and pepper. Discard the bay leaves, thyme, and parsley. Cover the vegetable with aluminum foil to keep warm. Meanwhile, make the gravy.
Skim off as much of the excess fat off the surface of the juices you poured into the saucepan. Pour the juices through a strainer to remove the solid pieces.
Place the saucepan on the stove over medium heat. Add the butter and slowly whisk in the flour to make a rue.
Slowly pour in the juices from the roasting pan and whisk continuously to avoid clumping. Once all the liquid has been added, turn up the heat to a boil. Let the gravy boil for about a minute to thicken. Be sure to stir it so it doesn't burn on the bottom of the pan.
Reduce the heat to low simmer for about 5 minutes.
Your overall cook time will be about 12 minutes per pound for medium-rare or 15 minutes per pound for medium.
I cook my roast at 450° for a short time to sear the outside and give it color. The length of time for this first part depends on the size. My guide is 5 pounds and under, 15-20 minutes. If it's 6 pounds or more, I cook it for 30 minutes for this first interval. I then reduce the heat to 350° and cook it at that temperature for the remainder of the cook time.
Here's a chart: