Preheat your oven to 450°F (232°C).

Season the prime rib roast generously with salt and freshly ground black pepper. Let it sit at room temperature for about 30 minutes.

In a large skillet over medium heat, heat the olive oil. Add the sliced leeks and cook, stirring occasionally, until they are caramelized and soft, about 15 minutes. Add the minced garlic and cook for an additional 2 minutes until fragrant. Remove from heat and stir in the chopped fresh tarragon. Set aside to cool slightly.

In a mixing bowl, combine the breadcrumbs, grated Parmesan cheese, and Dijon mustard. Add the caramelized leek mixture to the bowl and mix until well combined.

In a separate bowl, mix the softened butter, mashed black garlic, and lemon juice until smooth. Set aside.

Place the seasoned prime rib on a rack in a roasting pan. Spread the black garlic butter evenly over the top and sides of the roast.

Press the breadcrumb and leek mixture onto the top of the roast, ensuring it adheres well.

Roast the prime rib in the preheated oven for 20 minutes, then reduce the temperature to 325°F (163°C) and continue roasting until the internal temperature reaches your desired doneness, about 1.5 to 2 hours for medium-rare (135°F/57°C).

Remove the roast from the oven and tent it loosely with aluminum foil. Let it rest for at least 20 minutes before carving.

Serve the prime rib sliced, garnished with fresh tarragon sprigs, and enjoy with your favorite sides.