Preheat your oven to 350°F (175°C) and line a muffin pan with cupcake liners. In a large mixing bowl, whisk together the eggs, oil, applesauce, white sugar, brown sugar, and vanilla extract until well blended. Add the flour, baking soda, baking powder, salt, cinnamon, and nutmeg to the wet ingredients, mixing gently until just combined—do not overmix. Gently fold in the grated carrots and chopped pecans until evenly distributed throughout the batter. Use a large ice cream scoop to portion the batter evenly into the prepared muffin cups. Bake for 25 to 30 minutes, or until a toothpick inserted into the center of a cupcake comes out clean. Once baked, allow the cupcakes to cool in the pan for about 10 minutes before transferring them to a wire rack to cool completely.

To Prepare the Frosting: In a medium-sized bowl, combine the softened butter, cream cheese, powdered sugar, and maple butter. (If maple butter is unavailable, you may substitute with maple syrup.) Beat the mixture until it becomes smooth and creamy. Once the cupcakes are fully cooled, pipe or spread the maple cream cheese frosting on top and enjoy.