Preheat the oven to 350°F. Spray 2 (8-inch) round spring form cake pans with cooking spray.
In a small bowl, prepare the flax eggs by whisking together the ground flax and water. Set aside for at least 10 minutes.
In a medium bowl, mix the almond milk and vinegar. Whisk and set aside.
In a large bowl or stand mixer, mix the almond flour, chickpea or quinoa flour, coconut sugar, tapioca or arrowroot starch, cinnamon, baking powder, baking soda, salt, and nutmeg. Stir to blend.
To the bowl with the milk and vinegar, add the flax eggs, applesauce, and coconut nectar or maple syrup. Whisk until creamy.
Pour the wet ingredients into the bowl with the flour mixture, and while beating on medium speed, incorporate the melted coconut oil, continuing to beat until a batter is formed, increasing the speed as needed. Fold in the grated carrots and chopped nuts if using.
Evenly distribute the batter between the prepared cake pans, smoothing on top with a spatula until it reaches the sides. Bake for45 to 50 minutes until a toothpick inserted in the center comes out clean or the internal temperature reaches between 200° and 205°F. Allow the cakes to completely cool in the cake pans for about 2 hours.
Make the frosting. In a large bowl, mix the coconut oil and palm oil shortening or vegan butter. Beat with a hand mixer or immersion blender with the whisk tool until fluffy, about 1 minute.
Add the vegan cream cheese and beat again until blended. Add the tapioca or arrowroot starch and beat again. Add the maple syrup and vanilla and beat until everything is combined, using a spatula to scrape down the sides as you need to.
Using a spatula, mix the frosting. Frost the cooled cakes immediately or transfer the frosting to the refrigerator for 10 minutes for a denser texture. Spread one-third of the frosting on top of one cake. Place the second cake on top and add the remaining frosting on top and around the sides.