Preheat the oven to 350°F. In a medium bowl, combine the graham cracker crumbs, 1/4 cup granulated sugar, and melted butter. Press the mixture firmly into the bottom of a 9-inch springform pan to form an even crust.
In a separate large bowl, beat the softened cream cheese and 1/2 cup granulated sugar with an electric mixer until smooth. Add the eggs one at a time, mixing well after each addition, then stir in the vanilla extract. Pour the cheesecake batter over the prepared crust and set aside.
In another bowl, toss the chopped apples with brown sugar and cinnamon until evenly coated. Spread the apple mixture over the cheesecake batter in an even layer.
In a small bowl, combine the flour, rolled oats, and cold cubed butter. Use your fingers to pinch the mixture together until it forms coarse crumbs. Sprinkle the crumble topping evenly over the apples.
Bake in the preheated oven for 45-50 minutes, or until the cheesecake is set and the top is golden brown. Allow the cheesecake to cool completely in the pan before removing the sides and serving.