Preheat oven to 325° F. Grease a 9-inch springform pan with butter and set aside.

Use a food processor to crush graham crackers until fine crumbs form. Add sugar and melted butter. Pulse a few times until combined. Transfer mixture to prepared pan and firmly press the mixture to the bottom of the pan. I like to use the bottom of a measuring cup to press the mixture down.

Bake crust for 10 minutes. Remove from oven and set aside to cool. Place springform pan on top of aluminum foil and wrap the outside of the pan tightly with the foil. Make sure the foil comes almost all the way up the sides of the pan. If needed, do two layers of foil. Prepare cheesecake mixture.

Reduce the oven temperature to 300°F.

In the bowl of a stand mixer, beat cream cheese, sugar and brown sugar on medium-high speed until smooth, about 4 minutes. Scrape down the sides and bottom of the bowl as necessary.

Add the sour cream, vanilla and caramel sauce. Beat until combined, about 1 minute. Scrape down the sides and bottom of the bowl. One at a time, add the eggs and egg yolk, beating just until combined after each addition. Avoid over-mixing. Transfer batter to springform pan with graham cracker crust. Prepare water bath.

Fill a roasting pan with hot tap water until 1 inch high. Place roasting pan in the oven and then place the cheesecake in the water-filled roasting pan.

Bake at 300° F for 60 minutes. The cheesecake is done when the pan is gently shaken and the center of the cake jiggles slightly, but the edges are set. Turn the oven off, prop the oven door open and keep the cheesecake in the water bath for 1 hour.

Remove cake from waterbath and let it cool completely at room temperature before transferring to the refrigerator to cool for 4 hours or overnight.

Place apples, water, brown sugar, lemon juice, cinnamon, nutmeg in a heavy saucepan and stir over medium heat until the apples are tender, about 10 minutes.

Remove from heat and allow to cool. Spoon over the cheesecake and drizzle with caramel before serving.