Preheat the oven to 350° F.
Line a baking sheet with parchment paper, or with a Silpat baking mat. Set aside.
In the bowl of a stand mixer with a paddle attachment, beat the butter, brown sugar, and granulated sugar on medium speed until light and creamy, about 2 minutes.
Mix in the eggs, vanilla, and peppermint extract, scraping down the bowl as necessary.
In a separate medium bowl, whisk together the flour, cocoa, baking soda, and salt.
Combine the flour mixture into the wet ingredients, mixing just until combined.
Use a cookie scoop or tablespoon to scoop rounded heaps of dough onto the prepared baking sheet, leaving 2" of space in between each cookie.
Bake for 8 minutes.
While the cookies are baking, place the candy canes into a plastic food storage bag and crush them with a rolling pin.
Remove the cookies from the oven and quickly top with the marshmallow. Place back into the oven for an additional minute so that the marshmallow softens slightly and "melts" into the cookie.
Remove from the oven again and top with crushed peppermint candy canes. Allow the cookies to cool on a baking rack.
Store the cookies in a food storage container with a lid and they will stay soft and chewy for up to a week if they last that long.