Preheat oven to 350 degrees and place rack in center of oven. Place fluted paper liners in 18 muffin cups.
Toast the pecans or walnuts for about 8 minutes or until lightly browned and fragrant. Let cool and then chop coarsely.
Peel and finely grate the carrots. Set aside.
In a separate bowl whisk together the flour, baking soda, baking powder, salt, and ground cinnamon. Set aside.
In bowl of your electric mixer, with the paddle attachment, beat the eggs until frothy (about 1 minute). Gradually add the sugar and beat until the batter is thick and light colored (about 3 - 4 minutes). Add the oil in a steady stream and then beat in the vanilla extract and applesauce. Add the flour mixture and beat just until incorporated. With a large rubber spatula fold in the grated carrots and chopped nuts. Evenly divide the batter between the 18 muffin cups (fill each cup about 2/3 full) and bake 20 to 25 minutes or until a toothpick inserted in the center of the cupcake comes out clean.
Remove from the oven and let cool on a wire rack. With a knife or small metal spatula spread cream cheese frosting (recipe below) on top of each cupcake.
Cream Cheese Frosting
In bowl of electric mixer (or with a hand mixer), beat the cream cheese and butter, on low speed, just until blended with no lumps. Gradually add the sifted powdered sugar and beat, on low speed, until fully incorporated and smooth. Beat in the vanilla extract.
by The Savvy Kitchen