Preheat the oven to 350°F. Line two baking sheets with parchment paper.
In a stand mixer fitted with the paddle attachment, beat the butter until smooth and creamy, about 1–2 minutes.
Add the brown sugar and granulated sugar and beat until light and fluffy.
Add the egg, peppermint extract, and vanilla extract. Mix until fully combined.
In a separate bowl, whisk together the flour, cocoa powder, baking powder, baking soda, and salt.
Add the dry ingredients to the wet ingredients and mix on low speed just until a soft dough forms. Do not overmix.
Fold in the white chocolate chips.
Cover and chill the dough for 20–30 minutes.
Scoop the dough using a 2–3 tablespoon cookie scoop and place onto the prepared baking sheets, spacing a few inches apart.
Bake for 10–12 minutes, until the edges are set and the centers still look slightly soft.
Immediately after baking, gently press crushed candy canes onto the tops.
Let the cookies cool on the baking sheet for 10 minutes before transferring.