Line a baking sheet with waxed or parchment paper. In a mixer, cream the butter. Add the granulated sugar and beat until fluffy. Add the eggs, milk, and extracts and mix to combine. In a large bowl, combine the flours, baking powder, and salt and mix well. Add the dry ingredients to the sugar-egg mixture and mix to blend. Using an ice-cream scoop, scoop the dough onto the prepared pans. With a spatula, press and spread each cookie into a circle about 2 1/2 inches in diameter and about 3/8-inch thickness. Bake for about 15 to 20 minutes, until golden. Let cool on wire racks.
For the icing: In a medium bowl, stir the sugar, milk, and vanilla together until it forms a smooth icing. Transfer half of the icing to another bowl and stir in the cocoa powder and milk until smooth.
When cool, turn cookies over, so the flat side faces up. Spread white icing on half of each flat surface, then spread the other half with chocolate icing. Let set at room temperature for 30 minutes.