Preheat oven to 325°F (165°C) with rack in center. Grease a 9-inch springform pan and wrap the outside bottom and sides tightly with 2-3 layers of heavy-duty aluminum foil, coming up at least 3 inches.
Pulse 24 Oreo cookies in a food processor until fine crumbs form. Mix with 4 tablespoons melted butter until evenly moistened. Press firmly into the bottom and about 1 inch up the sides of the prepared pan, then freeze for 10 minutes.
Beat 32 oz softened cream cheese in a stand mixer fitted with paddle attachment on medium speed for 2-3 minutes until smooth. Add 1 cup sugar and beat for another 2 minutes until creamy.
Add 2 tablespoons flour and mix until just combined. Add 1 cup sour cream and mix until incorporated. Stir in 1 1/2 teaspoons peppermint extract and 1 teaspoon vanilla extract.
Add 4 room-temperature eggs one at a time, mixing on low speed after each addition just until the yolk disappears. Do not overmix. Pour the filling over the crust and smooth the top.
Place the foil-wrapped pan in a large roasting pan. Carefully pour boiling water into the roasting pan to reach about 1 inch up the outside of the springform pan. Bake for 60-70 minutes until edges are set and center has a slight jiggle.
Turn off the oven and prop the door open about 1 inch with a wooden spoon. Leave the cheesecake inside for 1 hour to cool gradually.
Remove the cheesecake from the water bath and pan, peel away the foil, and let cool completely at room temperature for 1-2 hours. Refrigerate for a minimum of 6 hours, preferably overnight.
Beat 1 1/2 cups cold heavy whipping cream on medium-high speed until soft peaks form. Add 3 tablespoons powdered sugar and 1/4 teaspoon peppermint extract and beat until stiff peaks form.
Run a knife around the edge of the pan, release the springform ring, and spread or pipe the whipped cream over the chilled cheesecake. Sprinkle with crushed candy canes just before serving.