Cream butter and powdered sugar in a bowl of a standing mixer until light and fluffy. Beat in vanilla.
Reduce the speed to low and beat in the flour mixture until just incorporated.
Gather dough and form a dish. Cover with plastic wrap and refrigerate for about 30 minutes to chill.
Preheat oven to 350F. Line 2 baking trays with parchment paper.
On a lightly floured surface, roll out the dough to 1/4 inch thickness. Cut into rounds using a floured cookie cutter. Place on the prepared baking sheets and place in the refrigerator for about 15 minutes.
Bake the cookies for 8 to 10 minutes, or until cookies are lightly browned.
Remove from oven and let the cookies cool a few minutes on the baking sheet before transferring them to a wire rack to cool completely.
To prepare icing, whisk powdered sugar, corn syrup, lemon juice, vanilla and 1 tablespoon water in a small bowl until smooth. Add more water if needed.
Transfer half the icing to another small bowl. Stir in cocoa powder. Thin with water if needed.
Spread white icing on half of each cookie’s flat side and cocoa icing on other half. Let stand until set, about 30 minutes.