Preheat oven to 400°F (204°C). Line two baking sheets with parchment paper and set aside.
Using a stand mixer with the paddle attachment, combine butter, brown sugar, and granulated sugar. Beat on medium speed until no visible butter chunks remain and the mixture is creamy.
Add the egg and egg yolk and mix until combined. Add the vanilla extract and peppermint extract. Scrape down the sides of the bowl and mix again until fully combined.
In a medium bowl, whisk together flour, cocoa powder, salt, baking powder, and baking soda.
With the mixer on low speed, gradually add the flour mixture to the wet ingredients. Mix until almost incorporated.
Add the semi-sweet chocolate chunks, milk chocolate chips, and white chocolate chips. Mix just until the dough comes together and the chocolate is evenly distributed.
Shape the dough into 8 large balls and place 4 on each prepared baking sheet, spacing them evenly apart.
Bake for 12-13 minutes, or until edges are set and the tops show slight cracks. The centers will look a little underbaked but will set as they cool.
Remove from the oven and immediately sprinkle the tops with crushed peppermint candies, pressing lightly. Carefully return cookies to the oven for about 1 minute to set the peppermint.
Remove from oven. Let the cookies cool on the baking sheet for 10 minutes, then transfer to a wire rack to cool completely.