Preheat oven to 375 degrees F (190 degrees C). Line baking sheets with parchment paper.
In a medium bowl, cream together butter and sugar until smooth. Beat in eggs one at a time, then set mixer on low speed, or stir in the milk, vanilla, and lemon extract. Combine cake flour and all-purpose flour, baking powder, and salt; gradually blend into the creamed mixture. Drop tablespoonfuls of the dough 2 inches apart on lined baking sheets. Put a small amount of water on a salad plate. Dip the bottom of a dry measure measuring cup into the water, then gently press down each cookie to flatten it. You will need to re-wet the measuring cup every 2-3 cookies or it will stick to the dough.
Bake until edges begin to brown, about 20 to 30 minutes (15-20 for smaller cookies). Cool completely.
Place confectioners sugar in large bowl. Mix in boiling water one tablespoon at a time until mixture is thick and spreadable. (Add more than the indicated amount if you need to).
Transfer half of the frosting to the top of a double boiler set over simmering water. Stir in the chocolate. Warm mixture, stirring frequently, until the chocolate melts. Remove from heat.
With a brush or the back of a spoon, coat half the cookie with chocolate frosting and the other half with the white frosting. Set on waxed paper until frosting hardens.