In a medium mixing bowl, combine the graham cracker crumbs and melted butter. Stir thoroughly until the crumbs are uniformly coated with butter.
Firmly press the crumb mixture into the bottom of a 9-inch springform pan to create an even and compact crust. Once done, place the pan in the refrigerator to chill while you prepare the cheesecake filling.
In a large mixing bowl, use an electric mixer to beat the softened cream cheese until creamy and smooth. Gradually add in the powdered sugar, vanilla extract, and peppermint extract; mix until the ingredients are fully incorporated and the mixture is silky.
In a separate bowl, whip the chilled heavy cream using a mixer until stiff peaks form, which should take about 2-3 minutes.
Carefully fold the whipped cream into the cream cheese mixture using a spatula. Be gentle to maintain the light and airy texture of the whipped cream.
Gently fold in the crushed candy canes, leaving a small amount aside for garnish.
Spoon the peppermint filling onto the chilled crust in the springform pan, smoothing the top with a spatula to ensure an even surface.
Cover the cheesecake with plastic wrap and refrigerate for at least 4 hours, or overnight for the best results, until set and firm.
When ready to serve, remove the cheesecake from the refrigerator. Top it generously with whipped cream and sprinkle the reserved crushed candy canes on top for a festive flair.
Notes
For an extra festive touch, serve the cheesecake with a drizzle of chocolate sauce and garnish with additional candy canes.