Preheat oven to 350 degrees F (175 degrees C). Grease and flour your baking pan.
In a large bowl, beat together eggs, oil, applesauce, sugar and vanilla extract.
Mix in flour, baking soda, baking powder, salt and cinnamon. Stir in grated carrots. Add walnuts. Pour into prepared pan.
Bake in the preheated oven for 45 to 55 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' sugar and 1 teaspoon vanilla. (I sometimes skip the frosting, flat plain cake is still delicious)
Beat until the mixture is smooth and creamy. Stir in chopped walnuts.
Frost the cooled cake. Enjoy with family!