Note--If using a flour mix that does not contain baking gum, add 1 tsp xanthan or guar gum. I use Bob's Red Mill GF Pizza crust, which does contain gum.
Cream sugar and eggs in a large mixing bowl with an electric beater or stand mixer. Add oil and vanilla and beat just until smooth.
In a separate bowl combine gluten-free flour mix, baking soda, baking powder, salt and cinnamon. Whisk to combine. Add the dry ingredients to the wet ingredients and beat until blended.
Stir in grated carrots, nuts, pineapple, and coconut. Pour the batter into prepared pans.
Bake in preheated oven for 45-55 minutes or until a toothpick inserted into the middle of the cake comes out clean. For muffins, reduce baking time to 30-35 minutes or until they pass the toothpick test. Cool on a wire rack.
While cake or muffins are cooling place butter, cream cheese and vanilla in a large mixing bowl and beat on high until smooth. Add powdered sugar and beat until smooth and creamy.