Preheat oven to 325 degrees. Lightly grease the doughnut pan. In a large bowl, mix flour, sugar, baking powder, nutmeg, cinnamon and salt. Stir in milk, eggs, vanilla and shortening. Beat together until well blended.
Fill each doughnut cup approximately 3/4 full. Bake 8 to 10 minutes in the preheated oven, until doughnuts spring back when touched. Allow to cool slightly before removing from pan.
To make glaze, blend confectioners sugar and hot water in a small bowl. Dip doughnuts in the glaze when serving. Recipe yields 12 spiced cake doughnuts.