Preheat oven to 350°F and line a cake pan with parchment paper.
In a large bowl, whisk eggs, applesauce, coconut oil, and vanilla until smooth.
In another bowl, combine almond flour, coconut flour, baking soda, baking powder, cinnamon, nutmeg, and salt.
Fold dry ingredients into the wet mixture, then stir in grated carrots, walnuts, and shredded coconut.
Pour batter into prepared pan and bake for 35-40 minutes, or until a toothpick comes out clean.
Cool completely before adding frosting if desired.