Prepare the Smoked Leek Compound Butter: In a skillet over medium heat, melt 2 tablespoons of butter. Add the chopped leek and cook until soft and slightly caramelized, about 5-7 minutes. Let cool. In a bowl, combine the softened butter, cooked leeks, smoked paprika, salt, and pepper. Mix well until combined. Transfer the mixture onto a piece of parchment paper and roll it into a log. Twist the ends to seal and refrigerate until firm.

Toast the Fennel Seeds: In a small skillet, toast the fennel seeds over medium heat until fragrant, about 2-3 minutes. Remove from heat and let cool, then crush slightly with a mortar and pestle or the back of a spoon.

Caramelize the Shallots: In the same skillet, add 2 tablespoons of olive oil over medium-low heat. Add the thinly sliced shallots and a pinch of salt. Cook slowly, stirring occasionally, until they turn golden brown and caramelized, about 15-20 minutes. Remove from heat.

Prepare the Prime Rib: Preheat your oven to 450°F (230°C). Rub the prime rib all over with the remaining 2 tablespoons of olive oil, and season generously with salt, pepper, and the toasted fennel seeds. Place the prime rib on a rack in a roasting pan. Spread the caramelized shallots evenly over the top of the roast.

Roast the Prime Rib: Roast in the preheated oven for 20 minutes. Then, reduce the oven temperature to 325°F (165°C) and continue to roast until the internal temperature reaches your desired doneness (about 130°F/54°C for medium-rare), approximately 1.5 to 2 hours. Remove the prime rib from the oven and tent with foil. Let it rest for at least 20 minutes before slicing.

Make the Balsamic Glaze: In a small saucepan, combine the beef broth and balsamic vinegar. Reduce over medium heat until thickened, about 10 minutes.

Serve: Slice the prime rib and serve with a dollop of smoked leek compound butter on top. Drizzle with the balsamic glaze and garnish with fresh tarragon.