Preheat the oven to 350 degrees F. Line an 8×8 square pan with parchment paper or grease the pan.
In a large bowl make the flax egg by combining the ground flaxseed and water. Mix and set aside for 5 minutes to thicken.
In a medium bowl, add the oat flour, almond flour, baking powder, baking soda, cinnamon, ginger, nutmeg, and salt. Whisk until well combined.
In the bowl with the flax egg, add the applesauce, oil, milk, vanilla, and coconut sugar. Whisk until well combined.
Add dry ingredients to the wet. Use a rubber spatula and mix until just combined (don’t overmix!). Fold in the carrots and walnuts. Pour batter into prepared pan. Bake for 35-38 minutes or until a toothpick inserted in the middle comes out clean. Let it cool in the pan for 10 minutes. Then lift cake out with parchment paper and place on a cooling rack to cool completely.
Once cake is cooled, frost with your favorite cream cheese frosting. Slice and enjoy! Keep leftovers in the fridge. It tastes even better the next day too!