1. Heat oven to 375 degrees F. Line 2 baking sheets with parchment paper.
2. Place butter and granulated sugar in a mixer fitted with a paddle and beat until light and fluffy. Add eggs, one at a time, mixing well between additions. Add milk and vanilla and beat well. Place flours and baking powder in a bowl and mix well. With mixer running, gradually add dry ingredients to butter mixture; beat until completely mixed.
3. Form the dough into tablespoon-size balls and place about 2 inches apart on the prepared cookie sheets. Transfer to the oven and bake until the edges of the cookies just begin to brown, about 25 minutes. Transfer to a wire rack and repeat with the remaining dough.Icing:
4. Place the confectioners' sugar in a large mixing bowl and add the water, 1 tablespoon at a time, until the mixture is spreadable but still thick. Transfer half the icing to another mixing bowl and stir in the melted chocolate. Set aside to cool.
5. Spread half of one side of the cookies with white icing and the other half with the chocolate icing. Set aside until the icing hardens completely.