Combine graham cracker crumbs, granulated sugar, and melted butter in a bowl. Press the mixture evenly into the bottom of a 9-inch springform pan, then bake at 350°F (175°C) for 8-10 minutes and let it cool.

In a large bowl, beat the softened cream cheese and granulated sugar with an electric mixer until smooth. Beat in the eggs one at a time, then stir in the sour cream, vanilla, and peppermint extract. Gently fold in the crushed candy canes, adding red food coloring if desired, without overmixing.

Wrap the bottom of your springform pan with aluminum foil for a water bath, then pour the cheesecake filling over the cooled crust. Place the springform pan in a larger roasting pan, filling it with hot water halfway up the sides. Bake at 325°F (160°C) for 60-75 minutes until the edges are set and the center jiggles slightly. Turn off the oven, let the cheesecake cool in the oven with the door ajar for 1 hour, then remove from the water bath and cool completely on a wire rack.

Heat the heavy cream in a small saucepan until it starts to simmer. Remove from heat and pour over the chocolate chips (and optional butter) in a heatproof bowl. Let it sit for 5 minutes, then whisk until smooth and glossy.

Once the cheesecake is completely cool and chilled, pour the chocolate ganache over the top, spreading it evenly. Decorate with additional crushed candy canes and chill the cheesecake for at least 4 hours, or preferably overnight, before serving.