Preheat your oven to 325°F (160°C).
Prepare the crust by mixing graham cracker crumbs and melted butter in a bowl.
Press the crumb mixture firmly into the bottom of a springform pan and bake for 10 minutes.
In a large mixing bowl, beat the softened cream cheese until smooth.
Add sugar and sour cream to the cream cheese and mix until well combined.
Add eggs, one at a time, beating well after each addition.
Stir in the vanilla extract and peppermint extract, mixing just until combined.
Fold in the crushed peppermint candies.
Pour the filling over the pre-baked crust and smooth the top with a spatula.
Bake for 50-60 minutes or until the cheesecake is set but still slightly jiggly in the center.
Turn off the oven and let the cheesecake cool in the oven with the door ajar for about an hour.
Transfer to the refrigerator and chill for at least 4 hours, ideally overnight.
Before serving, top with whipped cream and additional crushed peppermint candies, if desired.