Prep equipment: Preheat oven to 350ºF; line two 9" round cake pans with parchment paper, plus lightly grease. Set aside.
Make flax eggs: In a medium bowl, combine 1/4 cup flax seeds and 3/4 cup water and whisk together. Let seeds and water form gel, about 15 minutes total, whisking every 5 minutes.
Prep carrots: Grate carrots while flax seeds thicken. (I used my food processor with shredding disc + blade, see pictures in post)
Mix dry ingredients: In a large bowl add flour, baking powder, baking soda, salt, and spices. Whisk until evenly combined, then fold in grated carrots.
Mix wet ingredients: In the clean bowl of your food processor (or large bowl using hand mixer, or stand mixer with whisk attachment) combine sugars, flax "eggs", and vanilla extract until creamy, about 30 seconds. With the motor still running, add in applesauce, and pour steady stream of coconut oil until wet ingredients are evenly combined.
Make cake batter: Pour wet ingredients into large bowl of flour/carrot mixture. Using a large rubber spatula, gently combine and then add in milk to fully combine. Disperse batter evenly into prepared pans.
Bake: Bake cakes for 20-25 minutes, until toothpick inserted into center of each cake comes out clean. Let cake cool in pans for about 20 minutes before transferring to a wire rack to cool completely.
Make the frosting: Combine rinsed cashews with 3 tablespoons maple syrup and 1/3 cup of non-dairy milk in a high speed blender or food processor. Process until creamy, even consistency forms. Transfer frosting to a bowl and set in freezer for 30 minutes to firm.
Frost cake: Once chilled, whip frosting with hand mixer or use whisk. Spread a medium-thick layer of frosting evenly over first layer of cake, then layer second cake on top and frost completely. Store cake in fridge to chill and set frosting before serving.