Preheat oven to 350 degrees Fahrenheit. Spray an 8x8 pan with cooking spraying and line with two pieces of parchment paper, one in each direction to create flaps.
In a large bowl, combine applesauce, coconut oil, egg, and vanilla extract. Whisk until incorporated.
Add almond flour, oat flour, coconut sugar, baking soda, baking powder, salt, cinnamon, allspice, and nutmeg. Stir until batter is evenly combined. Transfer to prepared pan.
Smooth batter into an even layer with a rubber spatula, shake gently from to help even. Bake for 28-35 minutes, or until a toothpick comes out clean. Place pan on a wire rack until cool.
While the cake cools, clean out your bowl with soap and water (raw egg!) and combine frosting ingredients. Using an electric mixer, beat until mixture is smooth and creamy. Add more maple syrup if you desire a sweeter frosting.
Frost cake and sprinkle with additional cinnamon if desired. Store leftovers in an airtight container in the refrigerator.