In a medium bowl, whisk together the eggs, sugars, oil, apple sauce, and carrots until smooth. Pour into the dry ingredient mixture and stir together until just combined.
4 large eggs, 1 1/2 cups granulated sugar, 1/2 cup packed light brown sugar, 1 cup vegetable oil, 1/2 cup applesauce, 1 pound carrots
Fill the cupcake liners 3/4 full with the prepared batter. Then bake for 20 to 25 minutes or until a toothpick inserted into the middle of a cupcake comes out with just a few crumbs. I also recommend rotating the pan once in the middle of the bake time.
Cream Cheese Frosting
While the cupcakes are cooling, cream together the cream cheese and the butter in the bowl of a stand mixer. Mix in the powdered sugar one cup at a time. Then mix in the vanilla extract.
When the cupcakes are cool to the touch, spread the frosting over the top. Then top with pecan pieces. Please note that if you would like to pipe on the frosting as seen here, you will need to double the amount of frosting you make.
1/2 cup chopped pecans
Notes
Please note that if you are looking to pipe frosting onto the cupcakes as you see in the photos, you will want to double the frosting recipe to ensure you have enough.