Preheat oven to 350°. Line muffin pan with liners. Set aside.
Make the cake batter: In a large bowl, beat together egg, applesauce, oil, sugars, and vanilla. Mix in flour, baking soda, baking powder, salt, and cinnamon. Stir in carrots and nuts, if using. Pour evenly into 12 muffin liners (about 2 tablespoons per cupcake). Set aside.
Make the cheesecake batter: make sure all ingredients are room temperature to avoid lumps. Mix together cream cheese, sour cream, egg, sugar, and vanilla with a hand-held mixer. Pour evenly on top of cake batter in muffin pan (about 1-2 tablespoons per cupcake).
Bake for about 18-25 minutes, until cheesecake is no longer runny. Cool completely. Chill at least 1 hour before serving. You can frost them once they're room temperature, but don't serve until they've chilled a bit so the cheesecake can set.
Make the frosting: Using a hand mixer, cream butter and cream cheese until fluffy. Slowly add in powdered sugar and vanilla. Mix until smooth. Frosting option: use a 1M tip for a beautiful cupcake. Store in refrigerator for up to 3 days or freeze unfrosted cupcakes for up to 1 month.