Preheat oven to 350˚. Prepare a 9x13 pan.
In a large bowl, beat together eggs, oil, applesauce, brown and white sugars, drained pineapple and 2 teaspoons vanilla. Mix in flour, baking soda, baking powder, salt, nutmeg 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' sugar and 1 teaspoon vanilla. Beat until the mixture is smooth and creamy. Stir in chopped pecans. Frost the cooled cake.