Preheat the oven to 350°F. Line a 13x9-inch baking dish with parchment paper, leaving a two-inch overhang on the long sides for lifting out later.
In a stand mixer, beat the softened butter, brown sugar, and flour for the crust until crumbly and the mixture holds together when squeezed.
Press the crust mixture firmly and evenly into the bottom of the prepared pan.
Bake the crust for 14 to 16 minutes, until the edges are just beginning to brown. Leave the oven on.
While the crust bakes, make the filling. In a clean stand mixer bowl, whip the softened cream cheese on medium for about 1 minute until completely smooth.
Add the granulated sugar and beat until creamy, scraping the sides of the bowl at least twice.
Add the eggs and vanilla. Mix on low until just combined, scraping as needed. Don't overbeat.
Pour the cheesecake filling over the warm crust and smooth to the edges.
In a separate bowl, toss the diced Granny Smith apples with the sugar, cinnamon, and nutmeg. Sprinkle evenly over the cheesecake filling.
In the stand mixer, combine the softened butter, brown sugar, flour, and quick oats for the streusel. Mix until the topping clumps together but is not a paste.
Crumble the streusel evenly over the apple layer, breaking up large pieces with your fingers.
Bake for 40 to 45 minutes, until the topping is golden brown and the cheesecake center has only a slight jiggle when the pan is nudged.
Cool to room temperature on the counter, about 1 hour.
Refrigerate uncovered until cold, then cover and chill at least 4 hours total, ideally overnight.
Lift the chilled bars out of the pan using the parchment overhang. Slice with a sharp knife, wiping the blade clean between cuts.
Drizzle with caramel sauce just before serving.