Preheat the oven to 350 degrees Fahrenheit. Line a baking sheet with parchment paper.
In a medium bowl, whisk together the gluten free baking mix, baking soda and salt. Set aside.
Measure out ½ cup of milk. Add ½ tablespoon of white vinegar to the milk, stir, let sit 5 min and stir again. It will look chunky. Add the vanilla into the homemade buttermilk and stir until combined.
In a small bowl, using a spoon, stir together the cornstarch and water. It will be thick and tacky at first. Mix until watery and smooth.
In a large bowl, with an electric mixer, beat together the butter and sugar until soft and fluffy. About 2-3 min. Add the cornstarch water mixture and beat on low until combined. At low speed, beat in the gluten free flour mixture and the milk mixture, alternately in batches, beginning and ending with the gluten free flour mixture. Make sure to scrape down the sides of the bowl. Mix until smooth.
Using an ice cream scoop or ¼ cup, spoon batter onto prepared baking sheets making sure to leave 2 inches in between each cookie. Gently using your fingers press down the batter to make a dome instead of a ball. Gluten free flour has a way of staying exactly as it is placed down, so if you leave it in a ball shape, that’s how it will end up.
Bake for 15-17 min and immediately transfer to a cooling rack.
While the cookies cool make the icing. In a small bowl, stir together the confectioners sugar, corn syrup, lemon juice, vanilla and 1 tablespoon of water until smooth.
Take half the icing and place into a separate bowl. Stir in the cocoa powder and more water, ½ teaspoon at a time until it is the same consistency as the white icing.
Turn the cookies flat side up, then spread white icing over half the cookie and chocolate icing over the other half. Let the cookies dry on a cooling rack with wax paper underneath until hardened.