Using paper towels, blot the pumpkin puree to remove as much excess moisture as possible. Repeat as needed until the pumpkin is relatively dry. Set aside.
In a large bowl, whip the heavy cream until stiff peaks form, then set aside.
In another large mixing bowl, beat together the cream cheese, powdered sugar, brown sugar, pumpkin pie spice, and vanilla extract until the mixture is smooth and creamy. Pause occasionally to scrape down the sides and bottom of the bowl for a uniform texture.
Add the blotted pumpkin puree to the cream cheese mixture, beating until fully combined.
Gently fold the whipped cream into the pumpkin and cream cheese mixture with a spatula. Incorporate it carefully to maintain an airy texture, stirring until no streaks remain.
Retrieve the crust from the freezer and spoon the cheesecake filling over it, smoothing the top evenly.
Cover the cheesecake and refrigerate for at least 6 hours, or preferably overnight, to set fully.