Line a baking sheet with a piece of parchment paper.
Add the candy canes to a plastic bag and crush into small pieces using the back of a measuring spoon. Transfer them to a shallow bowl filled with 1 to 2 tablespoons of granulated sugar.
In a small bowl, whisk together the flour, cocoa powder, corn starch, baking powder, and salt.
In a medium mixing bowl, add the melted butter and sugars. Stir until combined and creamy. Add the egg yolk and vanilla extract and continue to mix until incorporated and smooth.
Sprinkle half of the dry ingredients over the wet ingredients and using a sturdy wooden spoon, mix until the dry ingredients are just incorporated. Add the remaining dry ingredients and mix until no dry streaks remain. Note - You can also mix in the dry ingredients using a hand held mixer - it is worth noting though that this is a tough and sticky dough and mixing by hand may actually be easier.
Using a cookie scoop (I use a smaller 1.5 tablespoon #50 cookie scoop), scoop 9 equal cookies. Roll the cookies between your hands to make dough balls. Roll the cookies through the peppermint sugar and place them at least 2 inches apart on the baking sheet. Using the back of a teaspoon, or your thumb, make wells in the middle of each cookie.
Cover the baking sheet lightly with plastic wrap and refrigerate the cookies on the baking sheet for at least 30 minutes. Note - You can also chill the dough first, then scoop, roll through peppermint sugar, and make wells after refrigeration. However, the cookie dough may be too firm to make wells without the cookies cracking too much.
With 15 minutes left of chill time, preheat the oven to 350F degrees and adjust the oven rack to the middle position.
Bake the cookies for 11 to 13 minutes, or until the edges are set. While baking the wells you made may fill back in slightly. Once the cookies come out of the oven, use the back of a teaspoon, or your thumb, to make indentations again. Cool the cookies on the baking sheet set on a wire cooling rack for 5 minutes, then transfer to the cooling rack to cool completely. Note - Even with forming the dough and chilling first, the cookies may crack a little around the edges when baked. This is ok! You can try to press the cracks back together as soon as they come out of the oven, or just leave them. The taste won't be compromised!