Preheat oven to 350°F. Spray a 9" springform pan well with nonstick cooking spray and set aside.
To make the crumb topping:
In a medium sized bowl, whisk together the flour, brown sugar, and cinnamon until well combined. Cut the cold butter into small pieces and using your hands or a pastry blender (a fork will work too), work it into the mixture until you have large clumps or crumbs.
Transfer to the refrigerator until ready to use.
To make the cream cheese filling:
In a medium sized bowl using a hand-held mixer, beat the cream cheese until smooth. Add in the sugar and mix until fully combined. Add in the egg and vanilla and mix until just combined. Set aside.
To make the cake:
In a large bowl, whisk together the flour, baking powder, baking soda, pumpkin pie spice, ground cinnamon, and salt. Set aside.
In a separate bowl, mix together the pumpkin puree, oil, brown sugar, granulated sugar, eggs, and vanilla until fully combined.
Add the dry ingredients to the wet ingredients and mix until just combined.
Pour half of the cake batter into the prepared springform pan and spread it around until the bottom of the pan is fully covered. Add all of the cheesecake filling on top and evenly spread it around. Then add the remaining cake batter on top and spread it around to cover the cheesecake filling.
Remove the crumb topping from the refrigerator and evenly distribute it on top.
Bake at 350°F for 45-55 minutes or until a wooden toothpick inserted into the center of the cake comes out clean.
Remove from the oven and place the pan on a wire rack to cool for about 15-20 minutes. Remove from the pan and transfer to the wire rack to finish cooling.
Notes
This is best eaten within a day or two, but can be stored in an airtight container in the refrigerator for up to 4 days. Reheat before serving, if desired.