Double Chocolate Cookies with a Peppermint Patty Surprise

Using an electric mixer, beat the flour, cocoa powder, both sugars, salt and baking soda on low speed. Beat in the butter, then the egg. Turn out the dough onto a large piece of plastic wrap and flatten into a disk; seal. Refrigerate until firm, about 1 hour.
Preheat the oven to 350 degrees F. Line 2 large cookie sheets with parchment. Working with 1 tablespoon of dough at a time, use your fingers to evenly encase each peppermint patty with dough. Place on the prepared pans and smooth the dough with your fingers. Bake, switching the pans halfway through, until just firm to the touch, about 15 minutes. Remove the pans from the oven and let cool slightly. Transfer the cookies on the parchment to a rack to cool completely.
Meanwhile, in a small bowl, microwave the white chocolate on high in 30-second bursts until creamy when stirred. Using a large spoon, drizzle the white chocolate over the cookies. Let cool completely before serving or storing.