Preheat the oven to 350 degrees. Arrange two racks evenly spaced in the oven.
Put the butter and granulated sugar in the bowl of an electric mixer fitted with the paddle attachment and beat on medium-high speed for 2 to 3 minutes, until light and fluffy. With the mixer on low, add the egg and vanilla and mix well, scraping down the sides with a rubber spatula. Sift the flour, baking powder, baking soda, and salt into a bowl. With the mixer on low, alternately add the flour mixture and sour cream in thirds and mix just until combined. Stir with a rubber spatula.
With a standard (2¼-inch) ice cream scoop, place 5 level scoops of batter 2 inches apart on each of two sheet pans lined with parchment paper. Bake for 10 minutes, then rotate and switch the pans from the top to bottom rack to bake the cookies evenly. Bake for another 6 to 8 minutes, until the edges are lightly browned and a cake tester comes out clean. Don’t overbake them! Cool on the pans for 5 minutes, then transfer to a wire rack set over a sheet pan to cool completely.
For the chocolate glaze, place the butter, chocolate, and coffee in a heatproof bowl and microwave on high for 30 seconds. Stir the mixture and continue to microwave in 30-second increments until the chocolate is almost melted, allowing the residual heat to finish melting the chocolate completely. Stir the mixture vigorously, until smooth. Turn the cookies so the flat side is up. Hold the cookie in your hand and, with a spoon, carefully pour the chocolate glaze on half of the cookie, forming a straight line down the middle. Allow the glaze to set for 30 minutes.
For the vanilla glaze, whisk together the confectioners’ sugar, corn syrup, vanilla, and 2 tablespoons of the cream, adding drops of cream until the glaze is smooth, thick, and barely pourable. Holding the cookie in your hand, with a spoon, pour the white glaze over the unglazed half of each cookie right up to the chocolate glaze. You can smooth it out with an offset spatula, if you need to. Allow to set for 30 minutes. Serve at room temperature.
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