Pipe a thin edge around the outside of each cookie, creating a “dam” for the icing.
Set aside 1/2 cup of the icing in an airtight container. With the remaining icing, add more water, 1 teaspoon at a time, until it is thin. (You want it thin enough so that if you dropped a spoonful back into the bowl, the icing would not hold its shape.)
Allow the cookies to dry for at least 6 hours. Once the cookies are completely dry, assemble the cookie cakes. Using the reserved icing, pipe a dot onto the center of one of the large circles.