Preheat oven to 325°F (160°C). Grease a 9-inch springform pan and wrap the outside tightly with aluminum foil to prevent water leakage.

In a medium bowl, mix graham cracker crumbs, sugar, and melted butter until combined. Press mixture firmly into the bottom of the prepared pan. Bake for 10 minutes, then set aside to cool.

In a large bowl, beat cream cheese and sugar until smooth and creamy. Add vanilla and mix well. Beat in eggs one at a time, mixing just until combined. Stir in sour cream until fully incorporated.

In a skillet over medium heat, combine diced apples, brown sugar, cinnamon, nutmeg, lemon juice, and butter. Cook for 8–10 minutes, stirring occasionally, until apples are tender and caramelized. Let cool slightly.

Pour half of the cheesecake batter over the cooled crust. Spoon half of the apple mixture over the batter, then gently swirl with a knife. Repeat with remaining batter and apples.

Place the springform pan inside a larger roasting pan. Fill the roasting pan with hot water until it reaches halfway up the sides of the springform pan (water bath).

Bake for 55–65 minutes, or until the edges are set and the center jiggles slightly. Turn off the oven, crack the door open, and let the cheesecake cool in the oven for 1 hour.

Remove from the water bath, discard foil, and refrigerate uncovered for at least 4 hours or preferably overnight before serving.