To make the cookies:
Position racks in the upper and lower thirds of the oven; preheat the oven to 375°F. Line two baking sheets with parchment paper or silicone baking mats.
In a small bowl, whisk together the all-purpose and cake flours, baking powder, and salt. In another small bowl, mix together the milk, 1 teaspoon vanilla, and lemon zest.
In a stand mixer fitted with the paddle attachment (or in a bowl by hand), beat together the butter and granulated sugar on medium speed until completely smooth. Beat in the eggs one at a time. Stir in half of the flour mixture, followed by the milk mixture, then stir in the remaining flour mixture and beat until the batter is smooth.
Drop 2 Tbs. batter in mounds spaced 2 inches apart on the prepared baking sheets. Bake, rotating the baking sheets midway through baking, until the cookies feel just set in the centers, about 15 minutes. Let cool completely on the baking sheets.
To make the icings:
In a medium bowl, whisk together the 2 cups confectioners' sugar with 2 tsp. of the corn syrup, the lemon juice, the vanilla, and 3 Tbs. water until smooth.
Transfer half of the mixture to a small bowl and whisk in the cocoa and remaining 2 tsp. corn syrup to make the "black" icing. Add up to 2 teaspoons more water, if necessary, to make the icing spreadable; it should not be too thin, so begin by adding 1 teaspoon and add another teaspoon only if needed. Whisk the remaining 2 Tbs. confectioners' sugar into the white icing. The two icings should have the same consistency: thick, but spreadable. (If the white icing is too thin, add a bit more confectioners' sugar.)
With a small icing spatula or a butter knife, spread white icing over one half of the flat (bottom) side of each cookie. Spread black icing over the other half. Let the icing set for a few minutes before serving.
Make Ahead Tips
The finished cookies can be stored for up to 2 days in an airtight container, preferably in a single layer so that the icing won't be marred.
Photo: Maren Caruso © 2010
Reprinted with permission from Ready for Dessert: My Best Recipes by David Lebovitz, copyright © 2010. Published by Ten Speed Press, a division of Random House, Inc.