Line 2 baking sheets with silicone liners.
Set to the side.
In the bowl of a stand mixer add the butter and sugar.
Using a paddle attachment, cream the butter and sugar on medium for 3 minutes. The butter/sugar mixture will appear light in color and fluffy.
Add one egg at a time, beating directly after adding.
After adding each egg, add the peppermint extract, and vanilla in the mixture.
Mix on low speed until combined.
Add in the red food coloring, on low speed until combined.
In a large bowl add both types of flour, cocoa, baking soda, cornstarch, and salt, whisk to combine. Slowly add the flour mixture to the butter mixture while the mixer is on low speed. Continue beating the mixture until a soft dough is made.
With the mixer on low beat in the white chocolate chips and crushed candy canes, mix on low until combined. Place the dough in the refrigerator, covered, until chilled for about 20 minutes.
Preheat the oven to 410 F.
Place the dough on a well floured surface.
Using a rolling pin, roll out the dough to a ½ inch thickness.
With a small sized circle cookie cutter, cut out 24 circles and place on a baking parchment covered baking pan. In a small bowl add the coarse sugar and crushed candy canes, mix to combine.
Dip each cookie into the sugar candy cane mixture and lightly press the mixture into the cookie.
Place on the baking sheet.
Bake for 10 minutes or until golden brown on the top.
Remove the cookies from the oven and let cool for 5 minutes.
Then transfer the cookies to wire racks to cool completely