Fit the work bowl of a stand mixer with the flat blade. Cream the cold butter for a few minutes until softened. Add the sugars and mix on medium speed for about 4 minutes. Stop and scrape the bowl as needed.
In a separate bowl, whisk together the flour, coconut, cornstarch and salt. Set aside.
Slowly pour the beaten egg down the side of the work bowl over the course of several minutes, mixing slowly but thoroughly. Add the vanilla or coconut extract, if using. Add the flour mixture in three additions, mixing well until the dough barely comes together. Turn the dough out onto a floured work surface and knead a few times to gather into a ball. Divide the dough in two, form into logs and wrap in plastic. Refrigerate for several hours or overnight.
Preheat the oven to 350 F. Line two baking sheets with parchment or a silpat for easier cleanup.
For the glaze, place the coconut in a small bowl. Cut the cold dough into thin slices, no more than 1/4 inch thick. Brush one side of each shortbread with the milk and dip in the coconut. Place each shortbread, coconut side up, about 1/2 inch apart on the cookie sheets. Bake for about 18 - 20 minutes or until well browned, rotating the cookie sheets if necessary.