Blitz gingersnap cookies into fine crumbs. Mix with sugar, ginger, cinnamon, and melted butter. Press firmly into a 9-inch or 10-inch springform pan, bottom and up the sides. Freeze 10 to 20 minutes.
Beat cold heavy cream on medium-high speed until stiff peaks form, about 4 minutes. Set aside.
Beat cream cheese until completely smooth. Add pumpkin puree, powdered sugar, and brown sugar and beat until combined. Add pumpkin pie spice, cinnamon, vanilla, and salt and beat until smooth and creamy with no lumps.
Using a silicone spatula, gently fold the whipped cream into the pumpkin mixture until just combined. Do not use the mixer.
Spoon filling into frozen crust. Smooth the top with an offset spatula.
Cover tightly and refrigerate at least 8 hours, preferably 12 hours or overnight.
Run a knife around the pan edge, release the ring, slice with a warm clean knife wiping between cuts. Top with whipped cream and caramel and serve cold.