Be sure to let the cookie logs cool after the first bake so they slice cleanly and don't crumble.
Preheat oven to 325°.
To prepare biscotti, place granulated sugar, molasses, 2 teaspoons vanilla, and 2 eggs in the bowl of a stand mixer; beat at high speed until ribbons fall from beaters (about 6 minutes).
Weigh or lightly spoon flours into dry measuring cups and spoons; level with a knife. Combine flours and next 5 ingredients (through baking powder) in a bowl; stir with a whisk. Add flour mixture to egg mixture; beat at low speed just until combined. Divide dough in half; place halves on a baking sheet lined with parchment paper. With lightly floured hands, shape each half into an 8 x 4-inch loaf. Lightly beat remaining egg in a bowl; brush tops and sides of loaves with egg. Bake at 325° for 35 minutes or until golden brown. Remove loaves from baking sheet, and cool 10 minutes on a wire rack.
Reduce oven temperature to 275°.
Cut each loaf diagonally into 11 (1/2-inch) slices; place slices, cut sides down, on baking sheet. Bake at 275° for 20 minutes, turning slices over after 10 minutes. Remove from pan, and cool completely on wire rack.
To prepare icing, combine powdered sugar, milk, and 1/4 teaspoon vanilla; stir with a whisk until smooth. Drizzle icing over biscotti; let stand until icing sets. Store biscotti in an airtight container.