Adjust the oven rack to the lower-middle position. Preheat oven to 350°F (180°C), or 160°C for a fan oven.
Prepare an 8x8 inch (20x20 cm) baking dish by lining it with parchment paper, allowing the parchment to hang over the edges slightly for easy removal.
Add the gingersnap cookies to a food processor and pulse until you get fine crumbs. Add the melted butter and brown sugar and pulse until combined.
Place the crumbs in the prepared pan and press them into the pan with the back of a spoon or the bottom of a measuring cup to form the crust.
Bake the crust for 7 minutes then remove and allow to cool on a wire rack while you make the filling.
Reduce the oven temperature to 325°F (165°C).
In the bowl of a stand mixer equipped with a paddle attachment, cream the cream cheese on high speed for 4-5 minutes until completely smooth and fluffy. Add the sugar and flour. Mix on medium speed until fully incorporated.
Add the sour cream, egg, egg yolk, and vanilla extract. Mix on medium-low speed until completely combined being careful not to overmix. Scrape the sides and bottom of the bowl and mix additionally if necessary.
In a separate bowl, combine the pumpkin puree, brown sugar, and heavy whipping cream. Whisk until combined and smooth.
Add the egg and vanilla and whisk until just combined. Add the flour and pumpkin pie spice. Stir until just combined and there are no dry spots remaining.
Set aside about ⅓ cup of the cheesecake filling for topping.
Pour the rest of the cream cheese mixture on top of the baked crust and use a spoon to smooth it out evenly.
Slowly pour the pumpkin mixture over the cheesecake layer and use a spoon to smooth it evenly. Use a clean spoon to place dollops of the reserved cheesecake mixture on top of the pumpkin layer. Then use a toothpick or the tip of a sharp knife to swirl the layers together.
Bake the bars for 50-55 minutes until the edges of the bars are set and the center only slightly jiggles.
Turn off the oven and crack the oven door to allow the cheesecake bars to cool slowly inside the oven for about 20 minutes.
Remove the pan from the oven and allow the bars to cool to room temperature and then transfer to the refrigerator to chill for 4 hours, or overnight. Slice into 16 squares and serve.