1-1/4 c. flour
1/2 tsp. baking soda
1/3 c. well-shaken buttermilk
1/3 c. (5-1/3 tbsp.) unsalted butter, softened
1/2 c. granulated sugar
1-1/2 cups powdered sugar
1 tbsp. light corn syrup
2 tsp. fresh lemon juice (grated zest is good, too!)
1/4 tsp. vanilla
1 to 2 tbsp. water
1/4 c. cocoa powder
Whisk together flour, baking soda, and salt in a bowl. Stir together buttermilk and vanilla in a cup.
Beat butter and sugar in a large bowl until pale and fluffy, about 3 min., 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 c. of batter about 2 inches apart onto a buttered large baking sheet or buttered parchment paper. 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 allow to cool.
Make icings while cookies chill:
Stir together powdered sugar, corn syrup, lemon juice, vanilla, and 1 tbsp. water in a small pan until smooth. Bring to boil and remove from heat. (this step helps to eliminate the raw sugar taste) Transfer half of icing to another bowl and stir in cocoa, adding more water, 1/2 tsp at a time, to thin to same consistency as white icing. Allow to cool a bit.
Turn cookies flat sides up, then spread white icing over half of each and chocolate over other half.