Heat the oven to 350° and grease a muffin tin or line it with wrappers.
In a medium bowl, combine the first seven ingredients.
In a large bowl, whisk together the oil, applesauce, eggs, and carrots. Slowly add the dry ingredients until just blended.
Divide the batter among the muffin cups. Bake the cupcakes until the tops are golden brown, about 25 minutes. Remove the pan from the oven and set it on a rack to cool.
Blend the cream cheese and maple syrup until smooth, then spoon the frosting into a zip-top bag. Snip the tip off one corner, and pipe frosting onto the cupcakes.