Make the crust: Combine graham cracker crumbs, brown sugar, and melted butter. Press into the bottom of the springform pan.
Prepare the cheesecake filling: Beat cream cheese and sugar until smooth. Add eggs, vanilla, pumpkin puree, cinnamon, and nutmeg. Mix until creamy.
Prepare the apple topping: Toss apple slices with sugar, cinnamon, and lemon juice. Set aside.
Make the crumble topping: Mix oats, flour, brown sugar, and cold butter. Rub together until crumbs form.
Assemble the cheesecake: Pour cheesecake mixture over crust. Layer apples on top and sprinkle with the crumble.
Bake: Preheat oven to 325°F (163°C). Bake for 50-60 minutes, or until set.
Cool and serve: Let cheesecake cool, refrigerate for at least 2 hours before serving.