Preheat your oven to 350°F (175°C). Wrap the bottom and sides of a 9-inch springform pan with heavy-duty aluminum foil to prevent water from seeping in during the water bath. Lightly grease the inside of the pan.

In a medium bowl, combine the graham cracker crumbs, 1/4 cup granulated sugar, and 6 tablespoons melted butter. Press the mixture firmly into the bottom of the prepared springform pan. Bake for 8-10 minutes, or until lightly golden. Remove from oven and let cool slightly while you prepare the fillings.

While the crust bakes, combine the sliced apples, 1/4 cup granulated sugar, lemon juice, and 1 teaspoon ground cinnamon in a medium saucepan. Cook over medium heat, stirring occasionally, for 8-10 minutes, or until the apples are slightly softened but still hold their shape. Remove from heat and set aside to cool completely.

In a large mixing bowl, using an electric hand mixer, beat the softened cream cheese and 1 cup granulated sugar on medium speed until smooth and creamy, about 2-3 minutes. Scrape down the sides of the bowl. Add the eggs one at a time, beating just until combined after each addition. Stir in the sour cream and vanilla extract until smooth. Be careful not to overmix.

Pour about half of the cheesecake batter over the cooled crust in the springform pan. Carefully spoon the cooled apple topping evenly over the cheesecake layer. Gently pour the remaining cheesecake batter over the apples, spreading it to cover the apples completely.

In a separate medium bowl, combine the all-purpose flour, rolled oats, light brown sugar, and 1/2 teaspoon ground cinnamon. Add the cold, diced butter and use your fingertips or a pastry blender to cut the butter into the dry ingredients until the mixture resembles coarse crumbs.

Sprinkle the crisp topping evenly over the cheesecake batter. Place the foil-wrapped springform pan into a large roasting pan. Carefully pour hot water into the roasting pan until it reaches about halfway up the sides of the springform pan. Bake for 55-65 minutes, or until the edges are set and the center still has a slight jiggle. The internal temperature should reach 150°F (65°C).

Carefully remove the springform pan from the water bath and place it on a wire rack. Run a thin knife around the edge of the cheesecake before it cools completely to prevent cracking. Let cool at room temperature for 1 hour, then transfer to the refrigerator and chill for at least 6 hours, or preferably overnight, before serving. Remove the springform rim just before serving.