In a bowl whisk together flour, baking soda, salt, and spices. In another bowl whisk together oil, sugar, eggs, and vanilla and with a wooden spoon beat in flour mixture until combined well. Add carrots, walnuts, and coconut, stirring until just combined. Fill muffin tins two thirds full with batter (there will be enough batter remaining to make 4 more cupcakes) and bake in middle of oven until a tester comes out clean, about 18 minutes. Turn cupcakes out onto a rack and cool completely. Muffin tins may be used again immediately for second cupcake batch. Cupcakes may be made 2 days ahead and kept in airtight containers.
03In a bowl beat together icing ingredients until fluffy. Icing may be made 2 days ahead and chilled in an airtight container. Soften icing at room temperature to spreading consistency.