Preheat oven to 350 degrees and place rack in center of oven. Place fluted paper liners in 18 muffin cups.

Toast the pecans or walnuts for about 8 minutes and until lightly browned and fragrant. Let cool and then chop coarsely.

Peel and finely grate the carrots and apple. Set aside.

In large bowl whisk together the flour, sugar, baking soda, baking powder, salt and cinnamon. Stir in the nuts and coconut. Set aside.

In separate bowl whisk together the eggs, oil and vanilla. Fold the wet ingredients, along with the grated carrot and apple, into the flour mixture, stirring just until moistened. Evenly divide the batter between the prepared muffins cups and bake for 20 - 25 minutes or until a toothpick inserted in the center comes out clean.

Remove from the oven and let cool on a wire rack. After about 10 minutes remove the muffins from the pans and cool completely on a wire rack before frosting.

Frosting
1/4 c unsalted butter, room temperature
4 oz. cream cheese, room temperature
3 tbsp. powder sugar
1/2 tsp vanilla

Beat the butter and cream cheese until very smooth with no lumps. Gradually beat in the powdered sugar until fully incorporated and smooth. Beat in the vanilla.