In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt. Set aside.
In a large mixing bowl, use a stand mixer or hand mixer to cream the softened unsalted butter and white granulated sugar together until light and fluffy, about 3-4 minutes.
Add the large egg and 2 1/2 teaspoons of pure vanilla extract to the butter mixture, mixing until fully combined.
Mix in the full-fat sour cream until the mixture is smooth and creamy.
Gradually add the dry flour mixture to the wet ingredients, mixing just until combined. Do not overmix! If the dough seems too sticky, add a little more flour, a tablespoon at a time, until it reaches a workable consistency.
Preheat your oven to 350°F (175°C). Using a cookie scoop or tablespoon, drop rounded balls of dough onto a parchment-lined baking sheet, spacing them about 2 inches apart. Gently flatten each ball to form a thick disk.
Bake in the preheated oven for 12-15 minutes or until the edges are lightly golden and the tops are set. Remove from the oven and let them cool on the baking sheet for a few minutes before transferring them to a cooling rack.
In a medium bowl, whisk together the sifted powdered sugar, 4 tablespoons of whole milk, light corn syrup, 1 teaspoon of vanilla extract, and 1/2 teaspoon of salt until smooth. Divide the icing into two bowls. In one bowl, stir in the cocoa powder until well combined for the chocolate icing.
Once the cookies are completely cool, spread the vanilla icing on one half of each cookie using an offset spatula. Then, spread the chocolate icing on the other half. Allow the icing to set for about 30 minutes before serving.
