Preheat the oven to 350 degrees F. Line 18 cups of 2 regular muffin tins with paper cupcake liners.
Sift the flour, granulated sugar, baking soda, cinnamon, and salt into a large bowl. In a separate bowl, whisk together the oil, eggs, vanilla and carrot puree. Stir the wet ingredients into the dry ingredients. Fold in the pecans and coconut.
Fill the prepared cups halfway. Bake until golden and a tester inserted in the middle of a cupcake comes out clean, about 30 minutes. Let stand for 5 minutes before popping the cupcakes out of the trays to cool completely on wire racks.
Frosting: Beat together the cream cheese, butter, confectioners' sugar, vanilla extract, and cinnamon, in a medium bowl. When the cupcakes are completely cool, slather the tops with a generous amount of frosting.
If you want to garnish the cupcakes, roll the edges of each cupcake the finely chopped pecans. Slice each dried apricot into 3 small carrot-shaped wedges. Arrange 2 apricot slices on each cupcake, along with mint sprigs to look like carrot tops. Store the cupcakes in an airtight container and keep refrigerated. They will keep for 3 to 4 days.