Prep Your Oven & Pan: Preheat your oven to 350°F (175°C). Line a 12-cup muffin pan with cupcake liners and set aside.
Make the Carrot Cake Batter: In a large bowl, gently whisk together the egg, applesauce, vegetable oil, brown sugar, granulated sugar, and vanilla until smooth. Next, fold in the flour, baking soda, baking powder, salt, and cinnamon. Finally, stir in the shredded carrots and optional nuts. Spoon this delicious batter evenly into your prepared muffin liners, about 2 tablespoons per cup.
Prepare the Creamy Cheesecake Topping: In a separate medium bowl, use a hand mixer to beat the softened cream cheese, room-temperature sour cream, sugar, vanilla, and egg until the mixture is completely smooth and free of lumps. Carefully spoon about 1-2 tablespoons of this cheesecake batter over the carrot cake batter in each muffin liner.
Bake to Perfection: Place the muffin pan in your preheated oven and bake for about 18-25 minutes. The cupcakes are done when the cheesecake topping is set and no longer jiggly. Let them cool completely at room temperature, then chill them in the refrigerator for at least 1 hour to ensure the cheesecake sets beautifully.
Whip Up the Frosting (Optional but Recommended!): While the cupcakes chill, prepare your frosting. In a bowl, cream together the softened butter and cream cheese with a hand mixer until light and fluffy. Gradually add the powdered sugar and vanilla extract, mixing until the frosting is smooth and wonderfully spreadable. Once the cupcakes are chilled, pipe or spread this luscious frosting on top.