Preheat oven to 350 degrees. In a mixing bowl, whisk together flour, baking soda, baking powder, salt, cinnamon, and nutmeg for 30 seconds, set aside. Shred carrots as directed in notes below, set aside.
In a separate large mixing bowl, whisk together granulated sugar and brown sugar, then add eggs and using an electric hand mixer, blend mixture on low speed 30 seconds, until slightly pale. Mix in applesauce and vanilla extract. With mixer running on low speed, slowly pour in vegetable oil and mix until combined, then mix 20 seconds longer. Mix in half of the dry ingredients, then pour the shredded carrots into the bowl with the remaining flour mixture and toss in flour mixture to coat carrots (they'll clump together a bit but that's fine). Pour carrot mixture into the batter and blend until evenly combined.
Pour batter into paper-lined muffin cups, filling each cup about 3/4 full. Bake in preheated oven until toothpick inserted into center of cupcake comes out clean, about 19 - 21 minutes. Cool in muffin tin several minutes then transfer to a wire rack and cool completely.