Preheat the oven to 350°F/180°C, and line 3 large cookie sheets with parchment paper or silicone baking mats.
In a large mixing bowl, combine the dry ingredients (flour, cocoa powder, baking soda, and salt). Set aside.
In the bowl of a stand mixer, cream the butter, and sugars for 3 minutes. Scrape off the sides of the bowl with a spatula. Then add in the egg, peppermint extract, and vanilla extract, and mix again until well combined.
Add the dry ingredients to the wet ingredients and mix until just combined (do not overmix the dough).
Divide the cookie dough into 1.5 tablespoon-sized balls using a medium-sized cookie scoop (each was 30 grams). Roll with your hands, and place on the prepared baking sheets.
Bake for 11-13 minutes. The cookies will puff up and have slightly cracked tops, make sure not to overbake them. Remove from the oven and cool for 5 minutes then transfer to a wire rack to cool completely.
In a small bowl, melt the white chocolate candy and dip half of each cookie shaking off any excess, then place on a wire rack or a parchment-lined sheet and immediately sprinkle with crushed candy canes. Allow to set completely.