Prepare Apple Filling: In a skillet, melt butter and cook sliced apples with apple cider, brown sugar, spices, lemon juice, cornstarch, and water until softened and thickened. Set aside to cool.
Make Crust: Combine graham cracker crumbs with melted butter. Press firmly into the bottom of a greased 9-inch springform pan. Chill while preparing cheesecake.
Prepare Cheesecake Batter: Beat cream cheese and sugar until smooth. Mix in eggs one at a time, then add vanilla, cornstarch, sour cream, and spices. Do not overmix.
Assemble Layers: Spread half the apple filling over the crust. Pour cheesecake batter on top.
Make Crumble Topping: Mix flour, oats, sugar, cinnamon, and salt. Cut in butter until crumbly. Sprinkle over cheesecake.
Bake: Place pan in a water bath and bake at 325°F (160°C) for 70–80 minutes, until center is almost set. Turn off oven and let cheesecake rest for 1 hour inside. Remove and cool completely. Refrigerate for at least 6 hours or overnight.
Prepare Caramel Sauce: Heat sugar in a saucepan until melted and amber in color. Stir in butter, then slowly add cream. Simmer briefly, stir in sea salt, and let cool.
Make Apple Topping: Cook apples with butter, cider, sugar, spices, lemon juice, cornstarch, and water until tender and glossy. Cool slightly.
Finish: Before serving, top chilled cheesecake with apple topping and drizzle generously with salted caramel.