Preheat your oven to 375° and position racks in upper and lower thirds. Line 2 cookie sheets with parchment or use silicone mats.
In a small bowl, combine flours, baking powder and salt. In another small bowl, mix the milk, vanilla and lemon zest.
Attach the paddle to your stand mixer (or use a bowl and wooden spoon) and cream the butter and sugar until smooth. Beat in eggs, one at a time. Stir in half the flour mixture, then the milk mix and then the reamining flour. Beat until smooth.
On pans, drop 2 tablespoons batter in mounds about 2 inches apart and flatten slightly.
Bake for about 15 minutes until the centers of the cookies feel just set. It is good to rotate the baking sheets halfway through baking.
Let cookies cool completely on the baking sheets.
Whisk together 2 cups of confectioner's sugar with 2 teaspoons corn syrup, lemon juice, vanilla and water until smooth.
Transfer half of this mixture to another bowl and whisk in the cocoa and remaining 2 teaspoons of corn syrup to make the black icing. The icing needs to be spreadable, so if necessary, add another teaspoon or two of water. Do not let the icing get runny.
Whisk the remaining 2 tablespoons of confectioner's sugar into the white icing. The two icings should have the same thick, spreadable consistency. If the white is too thin, add a little more sugar.
With a small spatula, spread white icing over flat side (bottom) of cookies. Spread the black icing over the other half. Let set for about 30 minutes before serving.