PREHEAT oven to 350ºF.
IN a mixing bowl, combine eggs, egg white, sugar, oil, and applesauce.
IN a separate bowl combine all dry ingredients. Add egg mixture to dry ingredients and beat all together.
ADD carrots, pineapple, and pecans (optional), and mix again. Spoon into 24 lined muffin cups; filling about 2/3 full.
PLACE pan on the middle oven rack and bake for 20 minutes, until a toothpick inserted into the middle comes out clean.
REMOVE from the oven and let cool for 5 minutes and then remove from the pan. Allow cupcakes to cool completely on wire racks before frosting.
IN a large mixing bowl cream together butter, cream cheese, and vanilla extract.
GRADUALLY beat in powdered sugar. Add milk to desired consistency. (Don’t make it too thin or it will slide off the cupcakes.)
FROST cupcakes with a small spatula or use your favorite pastry tip and bag. Garnish with crushed pecans, if desired.