Preheat the oven to 350 degrees F.

In the bowl of a stand mixer with the paddle attachment or using a hand mixer, cream the butter and sugar until the mixture is smooth and pale. Add the eggs, milk, vanilla extract and lemon extract and mix until fully incorporated.

Sift together the cake flour, all-purpose flour, baking powder and salt in a medium bowl. Add the dry ingredients to the wet ingredients and mix until just combined. With an ice-cream scoop, place portions of dough on a parchment-lined sheet pan about 3 inches apart. Using a spatula spread the dough into 3-1/2-inch diameter circles about 3/8-inch thick. Bake for 20 minutes until the edges are light golden brown. Cool completely on a wire rack.

For the icing, mix the confectioners’ sugar, 6 tablespoons milk and vanilla extract in a medium bowl until smooth. Turn the cookies, so the flat sides are facing up. Spread the vanilla icing on one half of all the cookies. Add the cocoa powder and milk to the remaining icing. Spread the chocolate icing on the other halves of the cookies. Allow them to set for 30 minutes at room temperature until the icing has hardened.

Store in an airtight container.