For the carrot-coconut birthday cake: Preheat the oven to 350 degrees F. Spray three 9-inch round cake pans with cooking spray and lightly coat them with coconut flour.
In a stand mixer fitted with the paddle attachment, mix together the coconut oil, coconut sugar, honey and eggs until creamy. (Don't worry if the mixture looks broken; the batter will become smooth as you add more ingredients.)
In a large bowl, whisk together the coconut flour, oat flour, cinnamon, baking soda, salt, ground ginger and baking powder. Add half of the dry ingredients to the wet ingredients and mix until evenly incorporated. On low speed, mix in the coconut milk until combined and then gradually add the remaining half of the dry ingredients. Turn the machine off and scrape down the sides of the bowl to ensure all the ingredients are incorporated. Fold in the carrots and applesauce. Divide the batter among the prepared cake pans.
Bake until the cakes begin to pull away from the sides of the pans and a toothpick inserted in the center comes out clean, 25 minutes. Let cool in the pans for 10 minutes.
Run a paring knife around the edge of the pans and then invert the cakes onto 2 racks. Let cool completely. (Do not move the cakes around too much; they are delicate.)
For the coconut cream cheese frosting: In a stand mixer fitted with the whisk attachment, whip the cream cheese, honey, powdered sugar and coconut cream on high speed until fluffy and evenly incorporated.
To assemble the cake: Place one cake layer on a 10-inch (or larger) cake stand. Scoop a quarter of the frosting on top and spread it to the edges. Stack another cake layer on top and spread with another quarter of the frosting. Stack the third cake layer and spread a third quarter of the frosting around the entire cake to serve as a crumb coat.
Refrigerate the cake for at least 30 minutes before frosting the entire cake with the remaining frosting. Sprinkle the coconut flakes over the top.