Preheat your oven to 350 °F (177 °C). Line the bottom of 2 8-inch Cake Pans with parchment paper and lightly grease the sides.
Combine the ground flax and water in a small bowl. Set aside for 10 min. It will thicken and gel up a bit.
In a medium mixing bowl, add the soy milk and apple cider vinegar. Combine and let it sit for 10 minutes to create vegan buttermilk. It will thicken and curdle slightly. Set aside. (soy milk curdles the best)
In a large mixing bowl with a Mesh Strainer on top (or use a Flour Sifter), sift together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt. Now whisk to combine well. Set aside.
In the medium bowl with the buttermilk, add the oil, applesauce, white and brown sugar, prepared flax egg, and vanilla. Whisk to combine well.
Now slowly pour the wet ingredients into the larger bowl of dry ingredients. Whisk together until just combined. Small lumps are fine. (*Don't over mix). Gently fold in the carrots to combine.
Divide batter between both pans and place in the oven for 35-40 minutes. To check for doneness, lightly press the surface - It should spring back (You can also use a toothpick and push it through the center - it should come out mostly clean).
Let the cake cool in the pan for 10-15 minutes before removing. Then let them cool on a rack completely before frosting. Frost and top with walnuts if desired. Enjoy!