Heat oven to 350°f.
Place parchment paper on 2 cookie sheets (jelly roll pan) and set aside.
Take the shredded zucchini and wrap it firmly in a tea towel, twist the towel to release the juice from the zucchini. When the zunnini is very dry, place in a bowl with the shredded carrot and set-aside.
In a large mixing bowl, combine the sugar and oil, mix on high speed, or medium speed if using a stand mixer, for about 3 minutes. Add the eggs, one at a time, and mix until incorporated. Mix in the vanilla. Reduce the speed to medium, or low, and add the dry ingredients, mixing until well blended and you have a fairly dense dough. Stir in the zucchini/carrot mixture along with the walnut pieces. Mix until incorporated.
Split the dough in two. Place each piece in the center of the parchment. Using very well-floured hands (the dough will be sticky) shape the dough into 2 evenly shaped logs (about 12 by 1/12 inches). Bake for 35 minutes; (the dough will spread as it cooks) remove to a rack to cool. When the logs are cool enough to handle, transfer to a cutting board and cut, crosswise and at a slight angle, into ½ inch slices. Lay the slices back onto the cookie sheet; return to the oven and bake an additional 15 - 20 minutes, or until the slices are firm when gently pressed. Transfer to cooling rack to cool completely.