In a mixing bowl, combine the softened unsalted butter, mashed black garlic, chopped parsley, lemon juice, salt, and pepper. Mix until well combined. Place the compound butter on a piece of plastic wrap, roll it into a log shape, and refrigerate until firm, about 1 hour.

Preheat your oven to 450°F (232°C). In a large cast-iron skillet or roasting pan, heat 2 tablespoons of olive oil over medium-high heat. Add the halved shallots to the skillet, cut side down, and sear until charred and caramelized, about 5-7 minutes. Remove from heat and set aside.

Pat the prime rib roast dry with paper towels. Rub the remaining 2 tablespoons of olive oil all over the roast. In a small bowl, combine the minced garlic, chopped wild thyme, kosher salt, black pepper, and smoked paprika. Rub this mixture generously over the entire surface of the prime rib.

Place the seasoned prime rib in the skillet or roasting pan, surrounded by the charred shallots. Roast in the preheated oven for 20 minutes to develop a crust. After 20 minutes, reduce the oven temperature to 325°F (163°C) and continue roasting until the internal temperature reaches your desired doneness (for medium-rare, aim for 130°F (54°C), about 1.5 to 2 hours).

Once the prime rib reaches your desired temperature, remove it from the oven and let it rest for at least 20 minutes before slicing. To serve, slice the prime rib and top each portion with a pat of black garlic compound butter.