Preheat oven to 350 degrees F (180 degrees C) and place rack in center of oven. Place paper liners in 18 muffin cups.
Toast the pecans or walnuts for about 8 minutes or until lightly browned and fragrant. Let cool and then chop coarsely.
Peel and finely grate the carrots and apple. Set aside.
In a large bowl whisk together the flour, sugar, baking soda, baking powder, salt, and ground cinnamon. Stir in the nuts and coconut. Set aside.
In a separate bowl whisk together the eggs, oil, and vanilla extract.
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 muffin cups and bake for 20 - 25 minutes or until a toothpick inserted in the center comes out clean.
Remove from 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.
Makes 18 standard-sized muffins.
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 extract.
Place a dollop of the cream cheese frosting on the top of each muffin.