In a large mixing bowl, whisk the flours, baking powder, baking soda, salt, and spices together.
In a separate bowl (or in your food processor like I did!) blend together the flax eggs, sugar, maple syrup, avocado oil, and apple cider vinegar for about 30 seconds to a minute until frothy.
Add the wet ingredients and carrot shreds to the bowl of dry ingredients and fold together to combine.
Bake in 9 inch cake pans at 350 degrees for 25-30 minutes, transfer to a cooling rack for about 10 minutes, and then remove from the pans to completely cool before frosting.
To make the cream cheese frosting, put all of the frosting ingredients in a medium mixing bowl, and beat with hand beaters until smooth.