Preheat your oven to 350°F (175°C).
Line your baking sheets with parchment paper to prevent sticking.
In a mixing bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt. Set aside.
In a separate large bowl, beat the softened butter and granulated sugar with an electric mixer on medium speed until light and fluffy, about 2-3 minutes.
Add the egg and 2 1/2 teaspoons pure vanilla extract to the butter-sugar mixture. Beat until well combined.
Mix in the full-fat sour cream until the batter is smooth and creamy.
Gradually add the dry flour mixture to the wet ingredients, mixing on low speed just until the dough comes together. Add more flour if dough is too sticky.
Drop rounded tablespoons of dough onto the prepared baking sheets, leaving space between each cookie.
Bake for 12-15 minutes, until edges are lightly golden but centers may look slightly underbaked.
Let the cookies cool on baking sheets for about 5 minutes, then transfer to a cooling rack to cool completely.
While cookies cool, prepare the icing: whisk together sifted powdered sugar, 4 tablespoons milk, light corn syrup, 1 teaspoon vanilla extract, and 1/2 teaspoon salt until smooth.
Divide icing into two bowls; add cocoa powder and remaining 3 tablespoons milk to one bowl to make chocolate icing.
Once cookies are completely cool, spread vanilla icing on half of each cookie and chocolate icing on the other half. Allow icing to set before serving.