Preheat oven to 350º F. Line 2 cookies baking sheets with parchment paper or silicone mats.
In a bowl stir together flours, cocoa, baking soda, and salt. Reserve.
In another large bowl mix together both sugars, avocado, applesauce, egg white, and vanilla. Mix until the sugars dissolved, about 3 minutes. Gradually fold in flour mixture with a spatula. Stir chocolate chips and walnuts, and stir to distribute evenly. Cover with a plastic wrap and refrigerate for 15 minutes.
Drop dough by tablespoonfuls onto prepared baking sheets and bake about 10 - 12 minutes. Cool on a baking rack.