1. Preheat the oven to 350°F and grease two 9-inch cake pans.

2. Mix the eggs and vanilla with an electric mixer until bubbly.

3. Add both sugars and beat until fluffy.

4. Mix in the oil and applesauce.

5. In a separate bowl, whisk together the flour, baking powder, baking soda, cinnamon, and nutmeg.

6. Fold the wet and dry ingredients together using a rubber spatula.

7. Fold in the grated carrots.

8. Divide the batter evenly between the prepared pans.

9. Bake 30–35 minutes, until a toothpick inserted in the center comes out clean; cool completely.

10. Beat the butter and cream cheese together for 2 minutes.

11. Add the confectioners’ sugar one cup at a time while mixing.

12. Add the milk and vanilla; beat 3 additional minutes until creamy.

13. Level the tops of both cake layers with a knife.

14. Place the first layer on a serving plate.

15. Spread 1 cup of frosting over the top layer.

16. Add the second layer, cut-side down.

17. Frost the top and sides of the cake.

18. Sprinkle crushed pecans on top.

19. Refrigerate at least 15 minutes before slicing.