For the cookies:
1 cup all-purpose flour
1 cup cake flour
1 tsp. baking powder
1/4 tsp. table salt
6 Tbs. whole or low-fat milk
1 tsp. vanilla extract
Grated zest of 1/2 lemon, preferably organic
1/2 cup (4 oz.) unsalted butter, at room temperature
2/3 cup granulated sugar
2 large eggs, at room temperature
For the icings:
2 cups plus 2 Tbs. powdered sugar, or more if needed
4 tsp. light corn syrup
1/2 tsp. freshly squeezed lemon juice
1 tsp. vanilla extract
3 Tbs. unsweetened Dutch-process cocoa powder
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.
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.