To make the cookies, cream together the butter and sugar in a large mixing bowl until fluffy. Add the eggs, milk, vanilla extract, and lemon extract. Mix until smooth.
In another bowl, combine the cake flour, all-purpose flour, baking powder, and salt.
Add the dry ingredients to the wet ingredients in batches, stirring well.
Using a soup spoon, drop spoonfuls of dough 2 inches apart onto buttered baking sheets and bake until the edges begin to brown, 20 to 30 minutes. Then remove the cookies from the oven and let them cool completely.
To make the frosting, put the powdered sugar in a bowl and stir in enough water to make the mixture thick and spreadable.
Transfer half of the frosting to the top of a double boiler over simmering water. Add the chocolate and warm the mixture until the chocolate has melted and the frosting is smooth. Remove from the heat.
With a brush, coat half of each cookie with chocolate frosting and the other half with white frosting.