While the cookie crust bakes and cools, make the cheesecake filling.
Whisk together 1 1/3 cups of sugar and 2 Tbsp of cornstarch in a small bowl. This helps prevent the cornstarch from clumping.
In a large mixing bowl or the bowl of a stand mixer fitted with a paddle attachment, beat 4, 8-oz packages of room temperature cream cheese and the sugar/cornstarch mixture on a low speed until the mixture is completely smooth. Scrape the sides and bottom of the bowl with a rubber spatula to make sure everything is properly combined.
Crack 4 eggs in a bowl and whisk them together. Gradually mix the eggs into the cheesecake batter in a few installments while mixing on a low speed. Then mix in 2/3 cup sour cream, 1/2 cup caramel (recipe above), and 1 tsp vanilla extract until just combined.
Pour the caramel cheesecake batter into the prepared pan. NOTE: If you use an 8-inch pan, it will be filled close to the top of the pan! But don't worry, cheesecake doesn't really rise as it bakes, so it shouldn't overflow in the oven. Double-check that the oven temperature has been lowered to 300°F/150°C, then place the cheesecake pan on a large cookie sheet and bake the cheesecake for 20 minutes.
Without opening the oven door, reduce the oven temperature to 225 F / 110 C and bake for an additional 2 hours. It sounds crazy, but low and slow is the best way to bake up a perfect, crack-free cheesecake that doesn't sink.
Once the cheesecake is done baking, turn the oven off and let the cheesecake sit in the oven for 1 hour without opening the door.
Next, remove the cheesecake from the oven and let it continue to cool at room temperature for a few hours. Once it reaches room temperature, cover the top of the cheesecake with plastic wrap and refrigerate the cheesecake for an additional 3-4 hours or overnight (preferred).