Preheat oven to 350 degrees F (175 degrees C). Grease and flour a 9x13 inch pan.
In a large bowl, beat together eggs, applesauce, granulated xylitol and 2 teaspoons 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 40 to 50 minutes, or until a toothpick inserted into the center of the cake comes out clean. Let cool in pan for 10 minutes, then turn out onto a wire rack and cool completely.
To Make Frosting: In a medium bowl, combine butter, cream cheese, confectioners' xylitol and 1 teaspoon vanilla. Beat until the mixture is smooth and creamy. Stir in chopped pecans. Frost the cooled cake.