Preheat your oven to 450°F (230°C).

In a dry skillet over medium heat, toast the coriander seeds and pink peppercorns until fragrant, about 2-3 minutes. Remove from heat and let cool slightly.

Using a spice grinder or mortar and pestle, coarsely grind the toasted coriander and pink peppercorns.

In a small bowl, combine the ground spices with kosher salt, black pepper, and 2 tablespoons of olive oil. Mix well to form a paste.

Pat the prime rib dry with paper towels. Rub the spice paste all over the roast, ensuring even coverage.

Place the halved shallots in a roasting pan and drizzle with the remaining 2 tablespoons of olive oil. Season with salt and pepper.

Place the seasoned prime rib on top of the shallots in the roasting pan.

Roast the prime rib in the preheated oven for 20 minutes, then reduce the temperature to 325°F (160°C). Continue roasting until the internal temperature reaches 130°F (54°C) for medium-rare, about 1.5 to 2 hours, depending on the size of the roast.

While the roast is cooking, prepare the roasted shallot butter. In a medium bowl, combine the softened butter, thyme leaves, minced garlic, and lemon zest. Mix until well combined and set aside.

Once the prime rib has reached the desired temperature, remove it from the oven and let it rest for 20-30 minutes before carving.

To serve, slice the prime rib and top each slice with a generous dollop of roasted shallot butter. Garnish with fresh herbs if desired.