Preheat your oven to 450°F (230°C). In a bowl, mix the roasted garlic, tarragon, Dijon mustard, olive oil, salt, pepper, rosemary, and thyme to create a paste.

Rub the garlic and herb paste all over the prime rib, ensuring it's evenly coated. Let it sit at room temperature for about 30 minutes.

Place the prime rib on a rack in a roasting pan. Pour the beef broth into the bottom of the pan. Roast in the preheated oven for 20 minutes, then reduce the temperature to 325°F (165°C). Continue roasting for approximately 1.5 to 2 hours, or until the internal temperature reaches your desired doneness (130°F for medium-rare).

While the roast is cooking, melt 1 tablespoon of butter in a skillet over medium heat. Add the chopped leeks and sauté until softened and slightly caramelized, about 8-10 minutes.

In a mixing bowl, combine the sautéed leeks with the remaining softened butter, smoked paprika, and salt. Mix well and set aside.

Once the prime rib is cooked to your liking, remove it from the oven and tent it with foil. Let it rest for at least 20 minutes before slicing.

Slice the prime rib and serve with a generous dollop of smoked leek butter on top.