Make the apple filling: In a saucepan over medium heat, melt the butter with the water, brown sugar, cinnamon, nutmeg, and lemon juice, stirring until the sugar dissolves. The source method also mentions vanilla here but gives no amount; if you want it, add about 3/4 teaspoon vanilla extract.

Add the sliced apples and stir gently to coat. Cover and cook over medium-low heat for about 10 minutes, until fork-tender. Stir in the cornstarch slurry and cook for another minute, until the juices thicken. Cool the apple filling completely.

Heat the oven to 350°F (180°C). Line the bottom of a 9-inch springform pan with parchment. Mix the Biscoff cookie crumbs with the melted butter until evenly moistened. Press firmly into the bottom of the pan and bake for 7–8 minutes. Cool to room temperature.

Reduce the oven temperature to 300°F (150°C). Make the cheesecake filling: Beat the room-temperature cream cheese, sugar, and vanilla on medium speed until smooth. Mix in the sour cream. Lightly whisk the eggs and yolk in a separate bowl, then mix them into the batter on low speed just until combined. Sift in the cornstarch and fold gently until smooth.

Make the crumble: Stir together the flour, oats, brown sugar, cinnamon, and salt. Add the melted butter and mix with a fork until chunky crumbs form.

Assemble: Pour half of the cheesecake batter over the cooled crust. Distribute the cooled apple filling evenly over it, then gently add the remaining batter and smooth the top. Sprinkle the crumble evenly over the cheesecake.

Bake at 300°F (150°C) for about 60–75 minutes, until the edges are set and the centre still has a slight jiggle. If the crumble browns too quickly, tent loosely with foil.

Turn off the oven, crack the door open, and let the cheesecake cool inside for 1–2 hours. Cool to room temperature, then refrigerate for at least 5 hours, preferably overnight, before removing the springform ring and slicing. Drizzle with caramel sauce if desired.