Preheat oven to 350°F.
Stir together flour, baking soda, and salt in a bowl and set aside.
Beat butter and sugar in a large bowl with a hand held electric mixer until pale and fluffy. Beat in the lemon zest and vanilla, then add egg, beating until combined.
Add the flour mixture and buttermilk mixture alternately by hand until you have a smooth batter.
Spoon 1/4 cups of batter about 2 inches apart onto a large baking sheet lined with parchment. Bake in middle of oven until tops are puffed and pale golden, and cookies spring back when touched, 15 to 17 minutes.
Transfer with a metal spatula to a rack and chill (to cool quickly), about 5 minutes (or let them cool at room temperature).
Make the Black & White Icing. Whisk the powdered sugar, milk, cream, corn syrup and vanilla together in a large bowl. Measure out about 1/3 cup and put it in a separate bowl
Add the cocoa powder and 1/2 teaspoon of water to the 1/3 cup that you measured out and stir until smooth. If the chocolate frosting seems too thick, thin it with a little water or milk.
Transfer the frosting to disposable decorating bags that do not have the tip snipped off. You won't need a decorating tip. When ready to use, snip off the tip of the decorating bags and pipe half moons of each flavor over the cookies.
Makes about 8 to 10 cookies (I usually get 8)