Preheat oven to 350°F (175°C). Line a baking sheet with parchment paper.
In a medium bowl, whisk together flour, baking soda, and salt.
In a large bowl, beat butter and sugar until light and fluffy. Add eggs one at a time, beating well after each addition. Mix in vanilla extract.
Gradually add the flour mixture to the butter mixture, alternating with milk, beginning and ending with the flour mixture. Mix until just combined.
Drop ¼-cup mounds of batter onto the prepared baking sheet, spacing them about 2 inches apart. Flatten each mound slightly with a spatula.
Bake for 15 minutes, or until the edges are lightly golden. Let cookies cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.
For the icing, in a medium bowl, whisk together powdered sugar, corn syrup, vanilla extract, and water until smooth. Transfer half of the icing to another bowl and stir in cocoa powder to make the chocolate icing.
Spread vanilla icing on one half of each cookie and chocolate icing on the other half. Let the icing set before serving.