In a large mixing bowl, mix together eggs, sugar, and brown sugar until blended with no brown sugar clumps. Add vegetable oil, apple sauce, vanilla extract, and carrots, continue mixing until blended.
Add all dry ingredients. Mix until fully combined.
Prepare two 9'' round pans, line with parchment paper at the bottom and lightly greased with butter, and lightly dusted with flour.
Pour equal amounts of batter into the two round pans. Let it set for 5 minutes to allow air bubbles to rise to the top and pop them with a toothpick.
Bake at 350 F for 20-25 minutes until toothpick insert comes out clean in the centre. One cake might be ready sooner as every cake bakes differently, so monitor both cakes.
Cool completely before icing.