Preheat the oven to 425°F. Spread the coarse sea salt evenly on the bottom of a roasting pan large enough to hold the potatoes. Place the potatoes over the salt and bake in the oven for 40 to 50 minutes or until easily pierced with a small knife. Every 15 minutes, give the potatoes a 1/4 turn to allow them to cook evenly and pick up the flavor of the sea salt. When done, remove the potatoes from the oven, brush any sea salt off the skins, and set aside to cool for 5 minutes. Reserve the salt to decorate the serving plates.

Cut off a 1/2-inch cap along the length of each potato using the flattest part of the potato as the cap. Scoop the potato flesh out of each cap into a bowl and discard the caps. Carefully scoop the potato flesh out of each potato bottom, leaving a 1/4-inch-thick shell. Place the potato flesh in a medium bowl and set the potato shells aside.
Add the 4 tablespoons softened butter to the potato flesh along with salt and pepper to taste, and 6 to 8 very thin white truffle shavings or twice as much black truffle, along with the truffle scraps. Mash with a large fork and taste for seasoning.

Refill each potato shell evenly with the potato and truffle mixture. Reheat the potatoes for 3 to 4 minutes in the oven or for 45 seconds in a microwave set on high.