Line 2 baking sheets with parchment paper. Set aside.
In the bowl of a stand mixer fitted with a paddle attachment, combine the sugar and butter on medium speed until fluffy. Add eggs, one at a time, allowing each egg to fully combine. Add the milk, vanilla and lemon extract, and mix until smooth.
In a medium bowl, combine the cake flour, all-purpose flour, baking powder and salt. Using a flexible rubber spatula, stir until well mixed. Working in batches, fold the flour mixture into the butter mixture until well combined.
Drop large spoonfuls of dough onto the baking sheets 2 inches apart. Bake the cookies until the edges brown slightly, about 18 minutes. Transfer the cookies to a wire rack and allow to cool completely.
In the top pot of a double boiler, add the confectioners sugar. Set aside.
In the bottom pot of a double boiler over high heat, boil 1 cup of water. Stir just enough boiling water into the sugar to make a thick icing. Add more sugar if the icing gets too thin.
Frost half of the flat side of each cookie. Return the pot with the white frosting to its position above the pot holding the water, and bring the double boiler to a simmer. Stir in the unsweetened chocolate and corn syrup until the mixture is smooth. To darken the frosting, add 1 or more tablespoons of cocoa as desired.
Frost the second half of each cookie. Allow the frosting to set, then store in an airtight container.
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