Preheat your oven to 350°F. Line a muffin tin with cupcake liners.

In a large bowl, beat egg, applesauce, oil, brown sugar, granulated sugar, and vanilla until smooth.

Add flour, baking soda, baking powder, salt, and cinnamon; mix until combined.

Stir in shredded carrots and nuts, if using.

Divide the batter evenly among the cupcake liners, about 2 tablespoons each.

Bake for 18-20 minutes, or until a toothpick inserted in the center comes out clean.

Let cupcakes cool completely before frosting.

Meanwhile, prepare the cream cheese frosting by beating softened cream cheese and butter until fluffy. Add powdered sugar and vanilla; mix until smooth.

Frost cooled cupcakes and serve.