Preheat the oven to 350 degrees. Line a 12-cup muffin pan with paper liners or coat with cooking spray, oil, or butter.
In a medium bowl, whisk together the flour, cinnamon, baking powder, baking soda, salt, and nutmeg. Set aside.
In a large bowl, whisk together the brown sugar, oil, applesauce, eggs, and vanilla. Whisk in the carrots. Add the dry ingredients to the carrot mixture and stir until just combined (stir just until the flour mixture is incorporated; do not over-mix).
Spoon the batter into the prepared pan, filling each cup about 3/4 full. Bake for 20 to 22 minutes, or until a wooden pick comes out clean or with little moist bits clinging to it.
Cool the cupcakes on a wire rack (cool completely before frosting; at least 1 hour).
Meanwhile, to prepare the frosting, in a large mixing bowl, beat together the cream cheese, butter, and vanilla. With the mixer on low speed, add the powdered sugar - one cup at a time - and mix until blended and creamy. Beat on high speed for 1 to 2 minutes, until light and fluffy. Cover with plastic and refrigerate until ready to frost the cupcakes (chilled frosting spreads better, especially if you plan to pipe it onto the cupcakes).
Spread or pipe the frosting over the cooled cupcakes.
Store leftover cupcakes in an airtight container in the refrigerator for up to 5 days.