Carrot Cupcakes: Preheat oven to 350 degrees F (180 degrees C) and place rack in center of oven. Lightly butter (or spray with a non stick vegetable spray) or line 12 muffin cups with paper liners.
In a large bowl whisk together the flour, baking soda, salt, and ground cinnamon. Fold in the chopped walnuts.
In another large bowl whisk the eggs until lightly beaten. Then whisk in the sugar, oil, and vanilla extract until slightly thickened. Fold in the applesauce and grated carrots. Then fold this mixture into the flour mixture until incorporated. Evenly fill the 12 muffin cups with the batter and bake about 20 - 23 minutes or until a toothpick inserted into the center of a cupcake comes out clean. Remove from the oven, place on a wire rack, and let cool completely before frosting.