Preheat the oven to 325°F. Grease a 9-inch springform pan and wrap the outside tightly with two layers of heavy-duty foil to prevent water seepage. Mix graham crumbs, sugar, melted butter, and salt until the mixture resembles wet sand. Press firmly into the bottom and about ½ inch up the sides of the pan. Bake for 10 minutes, then cool while you prepare the filling.

Beat softened cream cheese on medium speed until completely smooth and free of lumps, about 2 minutes. Add sugar and mix until combined. Scrape the bowl, then blend in sour cream and vanilla. Add eggs one at a time on low speed, mixing just until each disappears. Overmixing adds air that can cause cracks.

Pour the filling over the cooled crust and smooth the top. Place the springform pan inside a larger roasting pan. Pour hot water into the roasting pan until it reaches halfway up the sides of the springform. Bake for 55–60 minutes. The edges should look set while the center still has a slight jiggle, like gelatin. Turn off the oven, crack the door, and let the cheesecake cool inside for 1 hour.

While the cheesecake cools, melt butter in a skillet over medium heat. Add diced apples, brown sugar, cinnamon, lemon juice, and salt. Cook, stirring often, for 8–10 minutes until the apples turn tender and the juices thicken into a light syrup. Cool completely.

Remove the cheesecake from the water bath and run a thin knife around the edge. Cool to room temperature, then refrigerate uncovered for at least 4 hours or overnight. Before serving, spoon the cooled apple topping evenly over the cheesecake and drizzle with caramel sauce.