Preheat oven to 350°F (175°C) and lightly grease a 9-inch springform pan. Combine graham cracker crumbs, granulated sugar, melted butter, and peppermint extract until well mixed.

Press the mixture firmly into the prepared pan and bake for 8-10 minutes until lightly golden. Remove, cool completely, and reduce oven temperature to 325°F (160°C).

In a large bowl, beat softened cream cheese and granulated sugar until smooth and creamy. Scrape down the sides as needed.

Add unsweetened cocoa powder, vanilla extract, and peppermint extract, beating until just combined.

Add eggs one at a time, beating on low speed after each addition until just combined, being careful not to overmix.

Gently fold in sour cream, whole milk, and salt until the mixture is smooth and lump-free.

Pour the hot cocoa cheesecake batter over the cooled peppermint graham cracker crust.

In a microwave-safe bowl, melt marshmallows with milk or heavy cream until smooth, stirring halfway. Optionally, whisk in powdered sugar to thicken slightly.

Allow the marshmallow mixture to cool for a few minutes until warm but not hot, and slightly thickened.

Carefully place spoonfuls of the melted marshmallow mixture over the cheesecake batter and gently swirl with a skewer or butter knife to create a decorative pattern.

Prepare a water bath by wrapping the springform pan tightly with foil and placing it inside a larger roasting pan, then carefully pouring hot water halfway up the sides of the springform pan.

Bake the cheesecake for 60-75 minutes, or until the edges are set and the center is still slightly jiggly.

Turn off the oven, leave the cheesecake in the water bath with the oven door ajar for 1 hour to cool slowly and prevent cracks.

Carefully remove the cheesecake from the water bath and remove the foil. Let it cool completely on a wire rack at room temperature.

Once cooled, cover with plastic wrap and refrigerate for at least 8 hours, or preferably overnight, until thoroughly chilled and firm.

Once chilled, remove the sides of the springform pan and garnish with crushed candy canes, chocolate shavings, or whipped cream if desired.

Store leftovers in an airtight container in the refrigerator for 4-5 days. For longer storage, wrap tightly and freeze for up to 1-2 months, thawing in the refrigerator overnight.