Unwrap candy canes (if they are wrapped) and place them in a freezer-weight resealable bag. Using the flat side of a kitchen mallet or meat tenderizer, gently tap on the canes to break them into small pieces (do not over-do this or you will end up with only peppermint dust!). Separate the pieces from the dust, and set aside.
Preheat oven to 375 degrees. In a medium bowl, whisk together flour, cocoa powder, baking soda, and salt. Set aside.
Use an electric mixer to cream together butter and sugar until light and fluffy, about 2-3 minutes. Add egg, milk, vanilla, and peppermint extract, then beat until well combined. Add the flour and other dry ingredients from the first bowl a little at a time, and mix on low speed until combined.
Using a 2-tablespoon scoop, drop dough onto parchment-linked baking sheet. Arrange cookies evenly with plenty of room between them. Bake until they start to spread and become firm, 10 minutes.
Remove from oven, place four mini-marshmallows in the center of each cookie, and bake another 2 minutes, or until the marshmallows begin to melt. Transfer cookies to a wire rack to cool completely.
While the cookies are cooling, make the chocolate ganache drizzle: place semi-sweet chocolate chips in a heat-proof mixing bowl. Heat the heavy cream in a small saucepan until it is almost simmering, then pour it over the chocolate chips. Wait 3 minutes then whisk thoroughly and then allow the ganache to rest for about 5 minutes.
Using a fork, carefully drizzle the chocolate ganache across each cookie, leaving some of the marshmallow exposed. Test the ganache on a side plate to make sure it is ready, and if it's too thin, allow it to cool further before using.
Top each cookie with the candy cane pieces and enjoy!