The coating of chocolate adds just the right amount of sweetness to this crunchy, crisp biscuit.
Preheat oven to 375 degrees. Spread out nuts on a baking sheet; toast in oven, stirring occasionally, until fragrant and golden brown, about 8 minutes. Remove from oven; reduce temperature to 350 degrees.
Rub nuts between paper towels to remove loose skins; coarsely chop nuts.
Sift flour, baking powder, and salt together into a medium bowl; set aside. Put butter and granulated sugar in the bowl of an electric mixer fitted with the paddle attachment. Cream on medium speed until pale and fluffy. Mix in vanilla. Mix in eggs in two batches, mixing each until just combined. On low speed, mix in flour mixture until just combined. Stir in walnuts and apricots by hand.
Butter a large baking sheet; set aside. Transfer dough to a lightly floured surface; divide in half. With floured hands, shape each half into an 8 1/2-by-2 1/2-by-1-inch log. Place logs at least 2 inches apart on buttered baking sheet. Bake until golden, about 30 minutes. Transfer sheet to a wire rack; let logs stand 10 minutes.
Transfer logs to a cutting board. Clean and butter the large baking sheet; set aside. Brush logs with egg white, and sprinkle with sanding sugar. Cut each log crosswise on the diagonal into 12 pieces (each 1/2 inch thick). Lay flat on buttered sheet. Bake 7 minutes; turn pieces. Bake until golden, 7 to 9 minutes. Transfer to a wire rack; let cool until crisp.
Melt chocolate in a heatproof bowl set over a pan of simmering water, stirring occasionally, until smooth, 3 to 5 minutes. Let cool slightly.
Dip a flat side of 12 biscotti in melted chocolate, arranging cookies, chocolate sides up, on parchment paper as you work. Let stand until chocolate is set, about 30 minutes. Store in an airtight container, keeping chocolate-dipped ones in a single layer on top, up to 2 days.
Sanding sugar is used to coat the biscotti because it won't melt in the oven. To dip all the biscotti, double the amount of bittersweet chocolate.