Preheat your oven to 450°F (232°C).

In a small bowl, combine the smoked sea salt and black pepper. Rub the roast all over with olive oil, then evenly coat it with the salt and pepper mixture. Let it sit at room temperature for about 30 minutes to 1 hour.

While the meat is resting, heat a grill or grill pan over medium-high heat. Place the leeks cut-side down on the grill and char for about 4-5 minutes until they are nicely browned and tender. Remove from heat and set aside.

Place the seasoned prime rib on a roasting rack in a roasting pan. Insert a meat thermometer into the thickest part of the roast. Roast in the preheated oven for 20 minutes to develop a crust. Then, reduce the oven temperature to 325°F (163°C) and continue roasting until the internal temperature reaches your desired doneness (about 120°F for rare, 130°F for medium-rare, and 140°F for medium). This can take about 1.5 to 2 hours, depending on the size of the roast.

In a medium bowl, mix together the softened butter, chopped tarragon, chopped chives, Dijon mustard, and lemon juice until well combined. Taste and adjust seasoning if necessary.

Once the prime rib reaches the desired temperature, remove it from the oven and let it rest for at least 20-30 minutes. This helps the juices redistribute throughout the meat.

Slice the prime rib and serve it topped with charred leeks and a generous dollop of tarragon chive butter. Garnish with additional fresh herbs if desired.