Begin by preheating your oven to 375 degrees F. Line a baking sheet with parchment paper and set the sheet to the side.
In a large bowl, whisk together the cake flour, baking powder, salt, cinnamon, nutmeg, and cloves. Set aside.
Using an electric mixer, beat the butter and sugar together until light and fluffy. This should take about 3 minutes. Scrape down the bowl as needed.
Beat in the egg and vanilla and then the pumpkin puree. (The pumpkin puree will make the mixture look as if it's separated. This is normal.)
Beginning and ending with the flour mixture, alternately add the flour mixture in three additions and the milk in two additions at low speed until just combined.
Using a medium-size (5 cm) ice-cream scoop, scoop the dough onto the lined baking sheet. Moisten your fingers with a bit of water and gently press each mound of dough into a disc that's 3-inches in diameter and 1/2-inch thick. (You can also tap the baking sheet on the counter to help spread out the dough.)
Bake for 20 minutes or until the centers are firm and the edges just begin to brown. Cool the cookies on the baking sheet for 2 minutes. Then transfer them to a wire rack to cool completely.
In the meantime, prepare your icing.
In a double-broiler, melt the chocolate. Set aside.
In a medium bowl, whisk together the powdered sugar, corn syrup, and water. (If desired, thin the icing by adding more water, a little a at a time.)
Spoon half of the icing into a small bowl. Add the melted chocolate and stir to combine. (Again, add water to thin the icing if it seems too thick.)
Place two wire racks on top of parchment paper. Then use a small offset spatula to spread the white icing on one half of each cookie, allowing the excess to drip off the side. Once the white icing has set up for 15 minutes, repeat the process with the chocolate icing on the other half of each cookie. Allow the icing to harden for at least one hour.
Store the cookies in an airtight container. Separate the layers of cookies with sheets of parchment.