Preheat oven to 350˚F. Grease two 9-inch round cake pans OR a 13×9-inch baking pan.
Combine flour, pecans, baking powder, cinnamon, nutmeg and salt in a medium mixing bowl. Stir well and set aside. Whisk together eggs, oil, sugars, applesauce and vanilla. Stir in carrots until well blended. Add carrot mixture to flour mixture and stir with a spatula until combined.
Pour batter into prepared baking pan(s). Bake 20 to 25 minutes if using the round pans OR 30 to 35 minutes in the larger pan, or until wooden pick inserted in center comes out clean. Cool round cakes in pan for 10 minutes then turn out on wire rack to cool completely.
Beat frosting ingredients together until very smooth; spread onto cooled cake. For layer cake, place one cake on cake plate; spread frosting on top surface(only). Top with second layer and repeat (Sides of cake are not iced). Sprinkle cake top with additional chopped pecans, if desired. Serve immediately or cover and refrigerate.