Make the cookies: Position racks in the upper and lower thirds of the oven and preheat to 350 degrees F. Line 2 baking sheets with parchment paper. Sift the flour, baking powder, nutmeg and salt onto a large piece of parchment or into a bowl. Combine the milk and vanilla in a measuring cup. Beat the butter and granulated sugar in a large bowl with a mixer on medium speed until light and fluffy, about 2 minutes. Beat in the whole eggs and yolks, one at a time, then increase the speed to high and beat until combined, about 30 seconds.
Reduce the speed to low and add the flour mixture in 3 additions, alternating with the milk mixture, and beginning and ending with the flour. Increase the speed to high and beat 15 seconds to combine and lighten the batter.
Arrange mounds of batter (about 2 tablespoons each) about 2 inches apart on the prepared baking sheets. Bake, switching the pans halfway through, until the cookies are just golden brown around the edges, 13 to 15 minutes. Let cool 5 minutes on the baking sheets, then transfer to a rack to cool completely.
Make the icing: Beat the confectioners' sugar, corn syrup, lemon juice and vanilla in a large bowl with a mixer on low speed until combined. Increase the speed to medium and add 6 to 7 tablespoons cold water, a little at a time, until the icing is smooth and spreadable.
Transfer a little more than half of the icing to a medium bowl; tint orange with food coloring and set aside. Add the cocoa powder to the plain icing and mix on low speed until combined. Increase the speed to medium and add 1 to 2 tablespoons cold water, a little at a time, to make the chocolate icing the same consistency as the orange icing.
Spread icing on the flatter side of each cookie, covering half with orange and half with chocolate icing. Let sit until the icing hardens, about 1 hour.