Add 24 ounces (3-8 ounce boxes) of cream cheese to a mixing bowl. Using a paddle attachment and at slow speed, mix the cream cheese with 1 cup and 2 tablespoons of granulated sugar. The cream cheese should start to get creamier. Make sure to occasionally stop the mixer and scrape the sides of the bowl with a spatula so that everything is fully mixed.
After mixing the cream cheese for about 5 minutes, add 3½ tablespoons of cornstarch to the bowl. The mixture should be very creamy without lumps.
Add the eggs one at a time, again scraping the sides of the bowl with a spatula after each addition. Only mix until each egg is incorporated.
Pour the cream into the bowl. Only mix until the cream is incorporated, about 1 minute, then stop.
Cut open the vanilla bean and scrape out the seeds. Add the vanilla seeds and the lime zest to the cream cheese mixture.
Add 2 teaspoons of lime juice to the bowl. Mix just until everything is incorporated.
Pour the cheesecake batter into the prepared 9x13 sheet pan. Smooth the batter with an offset spatula so that the cheesecake is an even thickness.
Have a kettle of steaming hot water ready. Place the sheet pan on a larger sheet pan or in a shallow roasting pan on the middle rack in the oven. Pour the hot water into the sheet pan or roasting pan, surrounding the smaller pan. If you use a roasting pan, you should add 1-2 inches of water.
Bake the cheesecake for 30 minutes at 325 °. At the halfway point rotate the pan in the oven so that the cheesecake bakes evenly.
After 30 minutes remove the cheesecake from the oven. The center of the cake may appear to giggle and not be set. It will continue to set as the cheesecake cools off.
When the cheesecake is at room temperature, place it in the refrigerator for 1 hour.