Adjust oven rack to lower position and preheat oven to 200 degrees.
Pat the rib roast dry with paper towels and season it liberally on all sides with salt, pepper and rosemary.
Add all of the chopped vegetables into a large roasting pan. Pour in the wine and broth as well as the herbs to the roasting pan. Submerge the herbs in the liquid so they don’t scorch. Place a roasting rack inside the pan so that it sits firmly on the bottom of the pan. Place the seasoned roast on the rack with fat cap facing up.
Move the roasting pan to the oven and bake until center of roast registers 120 degrees (for medium rare) on an instant-read thermometer. This should take anywhere between 3 and 4 hours. For medium to medium-well, you can remove the roast from the oven when the internal temperature reaches 135 degrees. We really do not recommend cooking the roast much beyond this point, as it will become quite dry, but cook the roast to your family’s liking. When the roast has reached the perfect internal temperature, remove it from the oven, transfer it to a large plate and cover tightly with aluminum foil. Place it in a warm spot and allow it to rest for at least 10 minutes.
When you are about 20 minutes away from serving the prime rib, increase the oven temperature to 500 degrees (even higher if your oven is capable). Remove vegetables from the roasting pan and discard them. (Although, we really love mushy, salty carrots. If you do too, remove them to a serving bowl or start snacking!) Pour the juices from the roasting pan through a fine mesh strainer into a saucepan. Use a ladle to skim excess fat off of the top of the liquid. Bring the roasting liquid to a boil over high heat and immediately reduce the heat to low and simmer for an additional 2 minutes. Season the au jus to taste with salt and pepper if needed. Remove the saucepan from the heat and whisk in the chunks of butter one at a time until it is all incorporated. Keep the au jus in a warm spot, but do not bring to a boil again or the butter will separate and become oily.
Return the roasting rack to the empty pan. Remove the foil from the prime rib and place the roast back into the rack, once again with the fat cap on top. Place the roast back into the scorching hot oven and cook until the exterior is crispy and very brown, about 10 minutes. Remove the prime rib from the oven and allow it to rest 5 minutes.
Carve the prime rib into thick slices and serve immediately with plenty of that wonderful au jus that you made. Enjoy!
One Last Thing
If you have leftovers, make prime rib roast sandwiched with au jus the next day!