Preheat oven to 350 degrees F (175 degrees C). Grease and flour a 9 inch square pan. Sift together the flour, baking powder, baking soda, salt, cinnamon and nutmeg. Set aside.
In a large bowl, cream together the shortening, brown sugar and white sugar until light and fluffy. Beat in the eggs one at a time, then stir in the vanilla. Beat in the flour mixture alternately with the milk, mixing just until incorporated. Stir in the chopped apples.
Spread batter evenly in prepared pan. Bake in the preheated oven for 35 to 40 minutes, or until a toothpick inserted into the center of the cake comes out clean. Allow to cool.
To make the frosting: In a medium bowl, combine 1/4 cup shortening, confectioners' sugar, cinnamon and nutmeg. Beat until light and creamy. Beat in the milk, one tablespoon at a time, until desired spreading consistency is achieved. Spread over top of cooled cake.