Heat oven to 350°F. Grease cookie sheet; set aside.
Combine 2 cups flour, walnuts, baking powder, baking soda and salt in bowl; set aside.
Beat sugar and eggs in another bowl at medium-high speed until thick and lemon-colored. Add butter and vanilla. Continue beating until well mixed. Beat at low speed, gradually adding flour mixture and scraping bowl often, until well mixed.
Turn dough onto lightly floured surface (dough will be soft and sticky). Sprinkle dough lightly with 2 to 3 teaspoons flour; knead into dough. Shape into 2 (8x2-inch) logs with floured hands. Place 3 to 4 inches apart onto prepared cookie sheet; flatten tops slightly.
Combine egg white and water; brush over top of biscotti. Sprinkle with sugar. Bake 25-30 minutes or until lightly browned and firm to the touch. Let cool 15 minutes on cookie sheet.
Reduce oven temperature to 300°F.
Cut logs diagonally into 1/2-inch slices with serrated knife; arrange slices, cut-side down, on cookie sheet. Bake 16-20 minutes, turning once, or until golden brown. Remove to cooling rack; cool completely.
Place chocolate chips and shortening in 1-quart saucepan. Cook over low heat, stirring occasionally, 2-4 minutes or until smooth. Drizzle mixture over cooled biscotti.