Preheat an oven to 375°F In a large bowl, beat the butter, sugar, and lemon zest until well blended. Add the eggs, one at a time, beating well after each addition. Stir in the vanilla. In a small bowl, combine the flour, baking powder, and salt. Add to the butter mixture and blend thoroughly. Stir in the nuts. The dough will be soft.
On a lightly floured work surface, divide the dough in half. Lightly flour each piece and shape it into a log about 1 1/2" in diameter and 9" long. Place the logs about 3" apart on an ungreased baking sheet. Press each log down to make it about 3/4" thick and 3" wide. Bake until puffed and lightly browned on top, about 20 minutes.
Cool 10 minutes on the pan, then slide the logs onto a work surface. Using a long, sharp knife, cut each log crosswise into 3/4" thick slices. Make each cut with a single swipe of the blade. Don't use a sawing motion, which will break the cookies. Place the cookies, cut side down, on the baking sheet. (The cookies can be touching.) Bake 10 minutes.
Remove from the oven and, using tongs, turn each cookie over. Bake until the biscotti are golden, 10 minutes more. Transfer to wire racks to cool completely.
To make the icing: in a small bowl, combine the powdered sugar, lemon zest, and lemon juice and beat until smooth. Beat in additional drops of lemon juice if necessary to make an icing that will coat the biscotti lightly. Dip one end of each biscotti in the icing, turning to coat the tip evenly. Place on a wire rack until the icing sets.