Preheat oven to 350 degrees. Line two baking sheets with parchment paper. Sift flour, baking soda, and salt into a bowl; set aside. In the bowl of an electric mixer fitted with the paddle attachment, mix butter until creamy, about 2 minutes. Add granulated sugar; mix until fluffy, about 3 minutes. Mix in egg and 1/2 teaspoon vanilla. Mix in flour mixture in 3 batches, alternating with the buttermilk.
Roll tablespoons of dough into balls (or use a 1/2-ounce-capacity ice-cream scoop); place them 2 inches apart on lined sheets. Bake until bottoms turn golden, about 10 minutes. Transfer to wire racks; let cool.
Whisk confectioners' sugar, corn syrup, lemon juice, remaining 1/4 teaspoon vanilla, and 1 tablespoon water in a small bowl until smooth. If necessary, add more water (icing should be a bit thicker than honey). Transfer half to a small bowl. Stir in cocoa; if necessary, thin with water. Spread white icing on half of each cookie's flat side, and cocoa icing on other half; let set 30 minutes.
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