Preheat your oven to 350°F (175°C). Line two baking sheets with parchment paper.

In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt.

In a large bowl, beat the softened butter and granulated sugar until light and fluffy.

Add the egg and vanilla extract to the butter mixture, and mix until well combined.

Gradually add the dry ingredients to the wet ingredients, alternating with the milk, beginning and ending with the dry ingredients. Mix until just combined.

Drop 1/4-cup portions of dough onto the prepared baking sheets, spacing them about 2 inches apart.

Bake for 13 to 16 minutes, or until the edges are lightly golden. Allow cookies to cool on the baking sheets for a few minutes, then transfer to wire racks to cool completely.

In a medium bowl, whisk together the powdered sugar, corn syrup, 1/2 teaspoon vanilla extract, and 2 to 3 tablespoons of water until smooth and spreadable.

Transfer half of the icing to another bowl. Add the unsweetened cocoa powder and an additional 2 to 3 tablespoons of water to this bowl, mixing until smooth.

Turn the cooled cookies flat side up. Using a small spatula or knife, spread the vanilla icing on one half of each cookie.