In the bowl of an electric mixer, cream butter and sugar until light and fluffy, about 2-3 minutes. Add eggs, milk, and vanilla, blend thoroughly. Add flour, and baking powder and combine just until dough comes together. The dough may be very soft and sticky. Take the batter of the bowl, pat into a disk, wrap in plastic and place in the freezer for about an hour, or until firm enough to roll.
Preheat the oven to 350°. Line a baking sheet with parchment paper. On a floured surface roll the dough to about ¼ inch thick. Using a 2-in round cutter, cut out cookies and place on the baking sheet. Reshape dough and place in the freezer until you’re ready to bake the next batch. Bake the cookies for about 9 minutes or until the cookies are puffed and lightly brown on the edges. Let cookies cool in pan for 2 minutes and place on a cooling rack. Repeat with remaining dough.
make the icings:
Stir together confectioners sugar, corn syrup, vanilla, and 1 tablespoon water in a small bowl until smooth. Transfer half of the icing to another bowl and stir in cocoa, adding more water, ½ teaspoon at a time, to thin to same consistency as white icing.
ice the cookies:
Turn cookies flat sides up, then spread white icing over half of each and chocolate over other half.