For the dough: Add the granulated sugar and butter to the bowl of an electric mixer and beat until smooth. Add the vanilla and egg and beat again, scraping the sides of the bowl halfway through. Sift together the flour, baking powder and salt in a medium bowl. Alternate adding the dry ingredients and the milk to the mixer until the dough comes together. Flatten the dough into a disc and put it in a plastic bag. Refrigerate until chilled, about 2 hours.
Preheat the oven to 350 degrees F. Line 2 baking sheets with parchment paper or baking mats.
Roll out the dough to 1/4 inch thick and cut out as many cookies as you can in a variety of animal shapes. Place the cookies on the prepared baking sheets and bake until they're just barely done, 8 to 10 minutes; you want them to be set, but not brown. Let the cookies cool completely before decorating.
For the frosting: Sift the powdered sugar into a bowl, add the food coloring and 1/4 cup cold water and stir to combine
Transfer the cookies to a cooling rack set over a baking sheet and drizzle the frosting over them. Let the frosting dry before serving the cookies, about 2 hours.