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 . Line 2 large cookie sheets with parchment. Working with 1 tbsp. 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 double boiler or small, heatproof bowl set over a small saucepan of simmering water, melt half of the white chocolate, stirring until melted. Remove from the heat and stir in the remaining white chocolate until smooth. Using a large spoon, drizzle the white chocolate over the cookies. Sprinkle immediately with the crushed peppermint candies. Let set completely before serving or storing.