Position one of your oven racks in the center of your oven. Preheat oven to 350 degrees F. Line a baking sheet with parchment paper or a nonstick baking liner and set aside.
In a large bowl sift together the flour, cocoa powder, baking soda, and salt and stir in the sugar. Make a well in the middle of the dry mix and add two of the whole eggs, the egg yolks, and vanilla, blend the eggs a bit and mix into the dry mix until just combined. Add the nuts and chocolate chips and mix until incorporated. Don't worry if the dough seems too sticky; this is a very wet batter.
After mixing, scrape the dough out onto a lightly floured work surface and BRIEFLY knead it just until it comes together,adding water as needed. Divide it in half. Shape each half into a 2 1/2 inch by 12 inch flat log. The dough will be sticky so you may need to add more flour to your hands as you go along to accomplish this.
Carefully transfer the logs onto your prepared baking sheet, spacing them at least 3 inches apart because they will spread as they bake. Beat the remaining whole egg and brush it over the logs.
Place the baking sheet on the center rack of your oven and bake for about 35 minutes, or until the logs are firm to the touch.
Place the baking sheet on a wire rack and allow the logs to cool for 10-15 minutes.
Lower the oven temperature to 325 degrees F. Transfer the logs to a cutting board, discarding the parchment paper. Using a serrated knife, cut the logs on a slight diagonal into 1-inch thick slices. Place biscotti slices on the baking sheet cut side down.
Bake the biscotti until they are crisp, about 10 to 15 minutes turning halfway through baking.
Remove double chocolate biscotti from baking sheet and place on a wire rack and allow to cool completely.