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Cream the sugar and butter together in the bowl of an electric mixer fitted with the paddle attachment, just until combined. Add the eggs one at a time, scraping between each addition. Sift the flour, baking powder and salt together in a bowl and add to the butter and sugar mixture. Add the pistachios and apricots and mix to combine.
Dust a work surface with flour and roll the dough into a log. Transfer to the baking sheet and finish rolling the dough to the length of the baking sheet.
Bake the log of biscotti until it is light golden and springs back when lightly pressed, 25 to 30 minutes. Reduce the oven temperature to 315 degrees F. When the dough is cool enough to handle, slice into 3/4-inch slices and bake a second time until the biscotti have dried out and are golden brown, approximately 10 minutes.