For the cake: In a large bowl, combine gluten-free flour, baking soda, baking powder, cinnamon, nutmeg, clove, allspice, and salt. Mix until evenly incorporated and set aside. In a separate large bowl, whisk together eggs, sugar, and applesauce. Set aside. In another large bowl, combine carrots, pineapple, coconut, pecans, raisins, and vanilla. Mix and incorporate this mixture into the egg mixture. Gradually incorporate the dry ingredients into the wet ingredients and mix the batter until fully combined. Evenly distribute the batter into two 9x9 baking pans with the bottoms lined in lightly greased parchment paper. Bake in the oven preheated to 350 degrees Fahrenheit for about 35 minutes. Let cool completely on a wire rack.
For the frosting: In a stand mixer on medium setting, combine the butter and cream cheese until smooth. Slowly add in powdered sugar a little at a time until fully incorporated into the frosting. Next, add lemon juice and vanilla extract and continue to mix until combined. Once the frosting is fully mixed, you are ready to frost the carrot cake. For best results, serve cold and the next day.
