Fill a medium saucepan with potatoes, add water to cover, and salt. Bring to a boil, reduce heat, and simmer until tender, about 20 to 25 minutes. Meanwhile, trim black skin from truffle with a paring knife, and chop finely, saving the trimmings. Using a truffle slicer or Japanese mandoline, thinly slice the truffle, and reserve.
In a small saucepan, heat milk and unsalted butter over low heat until butter has melted. Drain potatoes into a colander. Put potatoes through a food mill, and transfer to an electric mixer fitted with a paddle attachment. On low speed, combine potatoes and milk mixture until no lumps remain; be careful not to beat any more than necessary. Add truffle butter, and combine.
In a small skillet over medium heat, saute leeks in 2 tablespoons olive oil until tender, about 2 to 3 minutes. Season with salt and pepper. Add reserved truffle trimmings, and saute for about 1 minute, until truffle is tender. Transfer leeks and truffles and 2 teaspoons chives to potatoes, and combine. Remove to a serving bowl, and sprinkle with remaining teaspoon chives. In the same skillet over medium-low heat, quickly saute the sliced truffles in remaining tablespoon of olive oil for about 1 minute. Garnish potatoes with sliced truffles, and serve immediately.
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