Combine the kosher salt, black pepper, 3 cloves of garlic, shallots, rosemary, thyme, and 1/2 cup of olive oil in the bowl of a food processor fitted with a steel blade. Pulse until well-combined. Set the roast bone-side down in a large baking dish and rub the marinade in with your hands. Cover with plastic wrap and refrigerate for at least 4 hours, and up to 24.

Remove the roast from the refrigerator about an hour before you plan to start cooking it to allow it to come to room temperature. Place the oven rack in the center of the oven, then preheat to 550 F. Insert a meat thermometer into the thickest part of the roast, and place (uncovered) in the oven. Reduce the temperature to 350 F right away. Cook about 2 1/2 hours (see note), until the thermometer reads 125 degrees. Stick the thermometer into a few other places - a range of 125-155 degrees will result in a nice range of doneness. Cover the roast with aluminum foil and set on the counter to rest for at least 15 minutes before carving.

While the roast is cooking, in a medium bowl, whip the heavy cream until stiff peaks form. Mix in the lemon juice, 1/2 teaspoon of salt, paprika, cayenne, and horseradish. Refrigerate until ready to serve.

Mince the last 2 cloves of garlic. Heat the butter and last 2 tablespoons of olive oil in a large skillet over high heat. When the butter is foamy, add the mushrooms, Worcestershire sauce, and minced garlic, toss to coat, and cook until the mushrooms are golden brown, 5-10 minutes. Remove from heat, and season with salt and pepper. Stir in the parsley. Transfer to a serving dish.

To carve the prime rib, slice the first piece close to the bone, between the ribs. Slice the second piece halfway between the next bone and continue this way to serve each person about half of the meat between the ribs. 

Serve topped with sauteed mushrooms and horseradish sauce.