Preheat your oven to 350°F (175°C). Line two large baking sheets with parchment paper or silicone baking mats to prevent the marshmallows from sticking to the pans.
In a large mixing bowl, beat the softened butter and granulated sugar together until the mixture is light and fluffy. This usually takes about 2 to 3 minutes with an electric mixer.
Add the egg, vanilla extract, and peppermint extract to the butter mixture. Beat on medium speed until fully combined and smooth.
In a separate bowl, whisk together the flour, cocoa powder, baking soda, and salt. Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined. Fold in the chocolate chips by hand.
Scoop about 1.5 tablespoons of dough and flatten it in your palm. Place half of a large marshmallow in the center. Fold the edges of the dough up and around the marshmallow, pinching it shut so no white is showing. Roll it into a ball.
Place the cookie balls on the prepared baking sheets, spacing them about 2 inches apart. Bake for 8 to 10 minutes. The cookies should look set on the edges but still slightly soft in the middle.
Immediately after removing the cookies from the oven, gently press the crushed candy canes into the tops of the soft cookies. If any marshmallow has leaked out, use a spoon to push it back toward the cookie while it is still warm.
Let the cookies cool on the baking sheet for at least 10 minutes before moving them to a wire rack. This allows the marshmallow center to firm up slightly so it doesn’t run everywhere.