Preheat oven to 325 degrees F. Grease a 9-inch springform pan.
Make the Cranberry Mixture: In a medium saucepan, whisk together the sugar and cornstarch. Whisk in the cranberry juice until smooth, then stir in the cranberries. Place the pan over medium heat and bring to a rolling boil. Cook and stir for 2 minutes, or until thickened. Remove from the heat.
Make the Crust: In a medium bowl, stir together the graham cracker crumbs and sugar. Drizzle over the melted butter and toss to combine, ensuring that all of the crumbs are moistened. Turn the crumbs into the prepared pan and press into an even layer in the bottom of the pan. Place the springform pan on a rimmed baking sheet. Bake for 10 minutes, then cool on a wire rack.
Make the Cheesecake Filling: With an electric mixer on medium-high speed, beat the cream cheese and sugar until smooth and fluffy, about 3 minutes. Add the flour and beat until combined, about 1 minute. Reduce the mixer speed to low and add the eggs one at a time, mixing until each is completely combined and scraping the sides of the bowl between each addition. Add the eggnog and vanilla extract and beat until just combined.
Pour two-thirds of the cheesecake batter into the prepared pan over the crust. Top with half of the cranberry mixture (cover and refrigerate the remaining cranberry mixture). Slowly pour the remaining cheesecake batter into the pan. Return the springform pan to the rimmed baking sheet and bake until the edges are set and the center jiggles slightly, about 1 hour 15 minutes.
Place the springform pan on a wire rack and cool for 10 minutes. Run a knife around the edge of the pan and cool for an additional 1 hour. Place the cheesecake in the refrigerator and chill overnight, at least 8 hours. When ready to serve, remove the sides of the pan and top with the remaining cranberry mixture. Leftovers should be stored in the refrigerator for up to 5 days.