For the Cookies : Preheat the oven to 375 degrees F.
Grease 2 baking sheets and line with parchment paper.
Sift both the flours, the baking powder, and salt together into a medium bowl. Then, sift the mixture again into another medium bowl.
In a large bowl with an electric mixer on medium, cream the butter and granulated sugar together until light yellow and creamy. Add the eggs, one at a time, beating well after each one. Beat in the extracts.
Sift about one-third of the flour mixture over the batter and, using a wooden spoon, stir in, then add about one-third of the cream and stir until mixed. Repeat until all the flour and cream is mixed in well.
Using a 1⁄4-cup ice cream scoop or 1/4 measuring cup scoop about 1⁄4 cup batter onto the prepared baking sheets for each cookie. Spread out with a small metal spatula into a 3-inch circle. Space the cookies about 3 inches apart to bake.
Bake just until the edges begin to turn light golden and the tops are puffed and spring back when touched, 12 to 13 minutes. The cookies should be only light golden on the bottom, not golden brown. A pick inserted into the center should come out clean. (Do not overbake!) Let the baked cookies cool for 5 minutes on the baking sheets, then transfer, upside down (bottom side up), to wire racks and cool completely.
For the Frosting: Sift confectioners' sugar into a medium bowl. Stir in the corn syrup, lemon juice, vanilla, and 1⁄2 cup hot water until smooth. Add more hot water, if needed, a little at a time, until the frosting is spreadable.
Transfer 11⁄4 cups frosting to another bowl; stir in the melted chocolate and the 3 tablespoons hot water.
Cover the frostings with a damp paper towel to keep them fresh and spreadable as you work; if the frostings stiffen too much, stir in a few more drops of hot water.
Using a small metal spatula,, spread white frosting over the entire flat bottom side of each cookie. When the frosting feels set, frost half of each cookie a second time, this time with chocolate frosting, layering it on top of the white and making the center line as straight as possible. Let the icing dry until it is no longer soft to the touch, at least 2 hours, and store the cookies, between sheets of parchment or waxed paper, in an airtight container at room temperature for up to 4 days. Do not refrigerate or freeze these cookies.