Begin by crushing the Lotus Biscoff cookies into fine crumbs using a food processor or a rolling pin in a resealable bag. Combine these crumbs with melted unsalted butter until the mixture resembles wet sand. Firmly press this mixture into the bottom of a lined 9-inch springform pan; it should create an even crust. Chill in the refrigerator while you prepare the other components.

In a mixing bowl, combine melted unsalted butter with all-purpose flour, brown sugar, and rolled oats. Stir until the mixture forms a thick, cohesive paste that clumps together. Allow the crumble mixture to cool slightly, setting it aside while you prepare the apple filling.

In a skillet over medium heat, sauté the sliced tart apples with brown sugar, ground cinnamon, and a little cornstarch until they soften and become syrupy, about 5-7 minutes. Stir gently to ensure the apples cook evenly and absorb the flavors. Once tender, remove from heat and let the apple mixture cool as you preheat the oven for the cheesecake.

While the apple filling cools, preheat your oven to 325°F (163°C). This temperature ensures that when you’re ready to bake the Apple Crumble Cheesecake, it will be at an optimal temperature for even cooking.

In a large mixing bowl, beat the softened cream cheese until smooth and creamy using an electric mixer. Gradually add granulated sugar, Greek yogurt, vanilla extract, and ground spices, mixing until fully incorporated. Slowly add in the large eggs one at a time, ensuring each egg is well blended before adding the next.

Pour the cheesecake batter over the chilled crust in the springform pan, smoothing the top with a spatula. Gently layer the sautéed apple mixture over the batter, followed by an even sprinkling of the crumble topping.

Place the springform pan on a baking sheet to catch any drips and bake in the preheated oven for 65-70 minutes. The Apple Crumble Cheesecake is ready when the edges are set but the center remains slightly jiggly. Keep an eye on it towards the end of baking to avoid over-baking.

Once baked, remove the cheesecake from the oven and let it cool in the pan for about an hour. After cooling, refrigerate the cheesecake for at least 6 hours, or preferably overnight, to set completely and develop flavors.