Preheat oven to 300 degrees F.In a food processor or a sealed plastic bag, finely crush about 24 of the Oreos to make 2 cups of crumbs.
Mix the crumbs with the melted butter and press tightly into a 9 inch round springform pan. Set the pan on a rimmed baking sheet (this will catch any grease) and bake for 10 minutes.
Let cool.In the bowl of a stand mixer or in a large mixing bowl using a handheld mixer, beat the cream cheese until creamy. Beat in the sugar, scraping sides until smooth. Using the lowest speed of the mixer, gradually add the egg yolks, scraping sides of bowl until mixed. On low speed, beat in the cream and vanilla.
Add the flour and stir on low until mixed, then add the vanilla.In a separate bowl using a handheld mixer, beat the egg whites until soft peaks form. Gradually add the extra 1/4 cup of sugar. Fold the egg white mixture into the cheesecake mixture.Break remaining Oreos into small pieces and fold into the batter.Scrape the batter into the pan and spread to the edges.
Bake at 300 degrees F. for 1 hour and 15 minutes.
Remove from oven and let cool 2 hours at room temperature, then chill overnight.
