Step 1Heat the oven to 375 degrees. Lightly grease and flour two cookie sheets, or line with parchment paper.
Step 2In a large mixing bowl, toss the flour, sugar, baking soda and salt with a fork to mix well.
Step 3In a separate bowl, beat the whole eggs with the vanilla just enough to mix yolks and whites. Stir the eggs into the dry mixture, then add the almonds and hazelnuts, gently kneading in the bowl to form a dough. Turn the dough onto a lightly floured board and continue kneading for a few minutes to distribute the nuts evenly throughout. Set the dough aside to rest for 10 minutes.
Step 4Divide the dough into 6 equal pieces and shape each into a log no more than 2 inches in diameter and three-fourths inch high. As each log is shaped, set it on a cookie sheet, keeping them a few inches apart to allow for expansion while baking.
Step 5In a small bowl, beat together the egg white with 1 teaspoon water to form an egg wash. Paint each log with the wash, then bake until the logs are lightly colored, 20 to 30 minutes. Remove from the oven and cool the logs on the cookie sheets. Meanwhile, reduce the oven temperature to 300 degrees.
Step 6When the logs are cool enough to handle, after about 20 minutes, cut them into half-inch straight slices. Stand them upright on the cookie sheet. (Leave a little space between the slices.) Return the biscotti to the oven for another 15 minutes.
Step 7Let the biscotti cool completely, before putting them in airtight containers or in the freezer.