Spread hazelnuts on a baking sheet and toast in oven 8 to 10 minutes, until fragrant. Let cool slightly. Place nuts in a dishtowel and rub together to remove some of the skin. Coarsely chop and set aside.
Place butter in the bowl of a stand mixer. Beat on medium speed until soft and creamy. Gradually add sugar and beat until light and fluffy, about 2 minutes. Add lemon rind and eggs, one at a time, beating well after each addition. Add vanilla and beat well.
Combine flour, baking powder and salt in a bowl. Stir to blend. Gradually add flour mixture to butter mixture, beating on low until a smooth dough forms. Stir in hazelnuts and apricots.
Divide dough in half. With lightly floured hands, form dough into two 10 x 2-inch logs. Place on an ungreased baking sheet, at least 2 inches apart. Bake 30 minutes, or until golden brown. Let cool on the baking sheet 15 minutes. Transfer logs to a cutting board. Using a sharp knife, cut straight down through the log to make 1-inch-thick slices; discard end pieces. Arrange on a baking sheet.
Bake 9 minutes or until lightly browned. Turn biscotti over and continue baking until lightly browned on the other side, about 9 minutes. Let cool completely on the baking sheets before serving or storing. The biscotti store well for one week and freeze well. Makes about 20.