Preheat oven to 350 degrees F (175 degrees C). Lightly grease and flour two 8 inch diameter cake pans.
Place the vinegar in a measuring cup, and pour in the milk to equal 3/4 cup. Set aside.
Beat the sugar with the shortening in a mixing bowl. Add the eggs one at a time, and continue beating until batter is smooth.
Combine the flour, cocoa, baking powder, cinnamon, nutmeg, and salt in another mixing bowl, and stir to blend. Stir the flour mixture into the egg mixture, alternating with the milk mixture. Stir in the lemon and vanilla extracts. Pour batter into prepared cake pans, dividing evenly.
Bake in preheated oven until a toothpick or small knife inserted in the center comes out clean, about 30 minutes. Cool 10 minutes. Run a small knife around the inside of the pans before inverting to remove the cakes. Allow cakes to cool completely before frosting.
To make the icing, beat the butter, egg yolk, and salt together in a mixing bowl. Gradually sift in the confectioners' sugar, alternating with the hot water, stirring until smooth. Do not add the cocoa until step 7.
To frost the cake, place one layer on a cake plate. Use about half the icing to cover sides and top of the cake. Stir the cocoa into the remaining icing and blend well. Place the second cake layer on top of the first, and frost with the chocolate icing, meeting the plain icing halfway down the side of the cake.