In a bowl of a stand mixer fitted with a paddle attachment (or in a bowl with an electric hand mixer), beat the cream cheese for 3 minutes until soft and fluffy. Add the sugar and brown sugar and beat for another 2 minutes. Scrape down the sides of the bowl and add the sour cream and vanilla, mixing just until combined. Add one egg at a time and mix until all are incorporated.
Remove 1 1/2 cups of cheesecake batter from the bowl and set aside for later.
To the initial bowl filled with cheesecake batter, mix in the pumpkin puree and pumpkin pie spice until no streaks remain.
Spoon half of the pumpkin batter into the pan and spread into an even layer. Dollop the top of the pumpkin batter with half of the plain cheesecake batter. Then, add the remaining pumpkin batter on top. Finally, dollop the remaining plain cheesecake batter on top.
Use a knife to swirl the various layers of cheesecake together.
Optionally, place the pan of cheesecake into a large round or rectangular pan, and fill the outer pan ⅓ of the way with boiling water to create a water bath. This will prevent the cheesecake from cracking in the center.
Bake for 50-55 minutes in the center rack of the oven, or until the cheesecake mixture is nearly set and has just the slightest jiggle in the center.
Turn the oven off and allow the cheesecake to sit in the oven for 20-30 minutes. Remove the cheesecake from the oven and let it cool on a wire rack for 1 hour. Then, transfer to the fridge to chill for 4-6 hours, or best overnight.
Use the parchment paper overhang to lift up the cheesecake and transfer it to a cutting board. Then, cut it into 16 squares.
Optional Garnish: Whip heavy cream with confectioners sugar and vanilla until stiff peaks form. Pipe whipped cream designs onto each cheesecake square and sprinkle with pumpkin pie spice.