i got this recipe from All recipes.com. the cake was super easy to make and extremely moist and tender. i have made alterations to the recipe as recommended by many of the members. Feel free to leave out the pecans. I did :)
Preheat oven to 350 degrees F (175 degrees C). Grease and flour a 9x9 inch pan.
In a large bowl, beat together eggs, oil, apple sauce, sugars, crushed pineapple and vanilla. Mix in flour, baking soda, baking powder, salt and cinnamon. Stir in carrots. Fold in pecans. Pour into prepared pan.
Bake in the preheated oven for 35-40 minutes, or until a toothpick inserted into the center of the cake comes out clean. Let cool in pan for 30 minutes.
To Make Frosting: In a medium bowl, combine butter, cream cheese, confectioners' sugar and vanilla. Beat until the mixture is smooth and creamy. Stir in chopped pecans. Frost the cooled cake.