Whisk together flour, baking soda, and salt in a bowl. Stir together buttermilk and vanilla in a cup.
Beat together butter and sugar in a large bowl with an electric mixer until pale and fluffy, about 3 minutes, then add egg, beating until combined well.
Mix in flour mixture and buttermilk mixture alternately in batches at low speed (scraping down side of bowl occasionally), beginning and ending with flour mixture.
Mix until smooth.
Spoon 1/4 cups of batter about 2 inches apart onto a buttered large baking sheet.
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.
Stir together confectioners sugar, corn syrup, lemon juice, vanilla, and 1 tablespoon water in a small bowl until smooth.
Transfer half of icing to another bowl and stir in cocoa, adding more water, 1/2 teaspoon at a time, to thin to same consistency as white icing.
Turn cookies flat sides up, then spread white icing over half of each and chocolate over other half.
•If you can stand the wait, cookies taste better if cooled without being chilled.
