Preheat oven to 350° F.
Cream butter and sugar in a large mixing bowl. Add cinnamon, nutmeg, vanilla, and salt.
Combine flour and baking soda; add milk alternately with flour and baking soda mixture, mixing until all ingredients are incorporated.
Stir in grated apple.
Pour into a greased 9x13 inch pan. Bake for 40-45 minutes, until toothpick inserted in the center comes out clean.
Place cake on a rack to cool.
Once cool, prepare frosting by combining butter, cream cheese, and vanilla in a large mixing bowl. Use a hand mixer to beat until smooth (about one minute). Add powdered sugar and beat until smooth, then beat 1-2 more minutes.