Preheat the oven to 350°F or 175°C and line an 8-inch square pan with parchment. Mix the crust ingredients and press them firmly into the bottom.
Bake the crust for 8 minutes, then remove it from the oven. Keep the oven on while preparing the remaining layers.
Beat the softened cream cheese and sugar until smooth. Add the eggs one at a time, then mix in the vanilla without beating excessively.
Pour the cheesecake mixture over the warm crust. Toss the apples with brown sugar and cinnamon, then scatter them evenly across the filling.
Rub the crumble ingredients together until clumps form, then sprinkle them over the apples. Bake for 35 to 40 minutes, until the center has only a slight wobble.
Cool completely at room temperature, then refrigerate for at least 4 hours. Lift from the pan and slice with a clean, sharp knife.
