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1
Make cookies:
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Preheat oven to 350F.
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3
Whisk together flour, baking soda, and salt in a bowl.
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4
Stir together buttermilk and vanilla in a cup.
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5
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.
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6
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.
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Mix until smooth.
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Spoon 1/4 cups of batter about 2 inches apart onto a buttered large baking sheet.
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Bake in middle of oven until tops are puffed and pale golden, and cookies spring back when touched, 15 to 17 minutes.
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Transfer with a metal spatula to a rack and chill (to cool quickly), about 5 minutes.
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11
Make icings while cookies chill:
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12
Stir together confectioners sugar, corn syrup, lemon juice, vanilla, and 1 tablespoon water in a small bowl until smooth.
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13
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.
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14
Ice cookies:
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15
Turn cookies flat sides up, then spread white icing over half of each and chocolate over other half.