Step 1 One or two days before you want to serve the potatoes, make truffle butter. Use a vegetable peeler or sharp paring knife to peel part of the truffle, then mince about half of it, and measure it. You want 3 tablespoons. If you don't have enough, mince a little more. Store the rest of the truffle until needed. Combine the truffle and the butter, mixing thoroughly, then roll it into a log in plastic wrap and refrigerate at least overnight.
Step 2 Scrub but don't peel the potatoes. Place them in a large saucepan, cover them with water by about an inch and add enough salt to the water to make it taste like the sea. Bring the water to a boil, then reduce heat to medium and cook the potatoes on a low boil until they're just tender when pierced with a knife, about 20 to 30 minutes. Bring the half-and-half to a simmer in a small pan over medium-high heat; once it simmers, turn the heat to very low.
Step 3Drain the potatoes and set aside to cool a little. Cut the cold truffle butter in small pieces. When the potatoes are cool enough to handle, peel them. Clean the pan, place it over low heat, and put the potatoes through a potato ricer into the pan. Add half of the hot half-and-half and stir, then add the butter a few pieces at a time, stirring to incorporate. Add additional half-and-half in stages with the butter, alternating as each is incorporated to achieve the desired consistency. Season to taste with additional salt. Serve immediately, shaving more truffle over each diner's portion at the table.