After pre-baking the crust, lower the oven temperature to 350°F.
In the bowl of a stand mixer fitted with the paddle attachment, add the 4 blocks of cream cheese. Mix on low-medium speed for about 1 minute, then add in 1 cup of sugar and 1 cup of sour cream.
Continue mixing on low-medium speed and add the eggs one at a time, mixing for a few seconds between each addition. Stop the mixer and scrape down the sides of the bowl.
Start the mixer again on low speed and add in ½ cup of half-and-half and 2 teaspoons of vanilla extract. Mix for another 30 seconds. Slowly add ⅓ cup of flour, then increase the speed to medium and mix for 2-3 minutes.
Scrape down the sides and bottom of the bowl to ensure everything is well incorporated. Pour the cheesecake batter into the prepared graham cracker crust.
Dollop ¼ cup of caramel sauce all over the cheesecake and swirl it with a knife to distribute the caramel throughout the batter.
Place a sheet tray or a 9x13-inch pan on the lower rack of your oven and fill it with about an inch of water to create steam. Put the cheesecake on the upper rack and bake for 45-55 minutes.
The cheesecake is done when the edges are golden brown and slightly pulling away from the sides of the pan, but the center is still jiggly. Turn off the oven and crack the oven door slightly. Let the cheesecake cool for 1 hour in the oven, then remove it and allow it to cool on the counter.
Once cooled, chill the cheesecake in the refrigerator for at least 4 hours or overnight. Once chilled, pour the remaining ½ cup of caramel sauce over the top and serve!