Preheat the oven to 350°F and line a cupcake tray with 12 cupcake liners.
Sift the all purpose flour into a mixing bowl and add the brown sugar, baking soda, baking powder, salt, cinnamon and nutmeg to a mixing bowl and mix them together.
Add the eggs, grated carrot, oil, applesauce, vanilla extract and vinegar and mix into a thick batter. If it seems like there isn't enough moisture for the batter to come together, then leave the bowl for a minute or two and then come back to it and mix again. The carrots will release more moisture into the batter and it will come together perfectly.*
Divide the batter among the 12 cupcake liners and place into the oven to bake for 25 minutes.
Let them cool for a few minutes before transferring to a wire cooling rack to cool completely before frosting.
Frost with a cream cheese frosting and decorate with chopped walnuts.