Preheat your oven to 350°F (175°C). Combine 2 cups of graham cracker crumbs with melted butter in a bowl until fully mixed, then press this mixture firmly into the bottom of a springform pan. Bake for about 10 minutes or until golden brown, then allow it to cool completely.
In a large mixing bowl, beat together softened cream cheese and granulated sugar until smooth and creamy. Add vanilla extract and eggs one at a time, mixing well after each addition until fully incorporated and fluffy.
In another saucepan over medium heat, combine sliced apples with lemon juice, cinnamon, and brown sugar. Cook until apples are tender but still hold their shape; this should take about 5-7 minutes.
Pour half of the cheesecake batter over the cooled crust, followed by an even layer of cooked apples. Add the remaining cheesecake batter on top, smoothing it out gently.
In a small bowl, mix remaining graham cracker crumbs with brown sugar and melted butter to create your crisp topping. Sprinkle this mixture evenly over the top layer of cheesecake before placing it in the oven.
Bake for approximately 60-70 minutes until set but slightly jiggly in the center. Once baked, allow it to cool at room temperature before refrigerating for at least four hours or overnight for best results before slicing and serving.