Add the cream cheese, sugar and corn starch to a large mixing bowl (or to a stand mixer fitted with the paddle attachment.) Beat the mixture until it's smooth, for about 2 minutes.
Add both eggs into the cream cheese, and beat the mixture on low speed *just* until the eggs have been fully incorporated.
Add in the heavy cream and vanilla, and again beat on low speed just until the mixture is cohesive.
Remove 1 ¼ c. of the cheesecake batter and place it in another small bowl. To the small bowl, carefully mix in the pumpkin puree and pumpkin pie spice just until the mixture is cohesive and no streaks remain.
Alternate pouring the vanilla and pumpkin cheesecake batters over the shortbread crust. To make the swirl on top, make sure to leave just a little bit of pumpkin batter and top the cheesecake off with a smooth layer of the vanilla cheesecake. Place dollops of the remaining pumpkin cheesecake on top of the vanilla, then use a knife to gently swirl the batters together.
Tap the pan on the counter a few times to help remove any air from the filling.
Fill a 9x13" baking dish, or a 10" cake pan a third of the way with hot water. Wrap tinfoil tightly around the bottom of your springform pan to avoid the possibility of water seeping into the springform pan. Place the springform pan into the larger pan with water.
Place the cheesecake with the pan in the oven and bake for 60-70 minutes, or until the cheesecake only slightly wobbles and reaches an internal temperature of 160°F / 71°C.
Once the cheesecake is fully cooked, turn off the oven and leave the door slightly ajar with the cheesecake in it for 1 hr. before transferring it to the fridge to chill for at least 8 hours.