Preheat oven to 350 degrees F (175 degrees C). Grease a 9-inch pie plate.
Mix oats, confectioners' sugar, butter, shortening, cinnamon, and salt in a bowl until the dry ingredients are entirely moistened; press evenly into the bottom and up the sides of the prepared pie plate.
Bake in the preheated oven until beginning to brown, about 20 minutes. Cool for at least 10 minutes.
Beat brown sugar, cream cheese, pumpkin puree, white sugar, and pumpkin pie spice together with a mixer until smooth. Fold whipped topping into the pumpkin mixture until smooth; pour into the pie crust.
Refrigerate pie until filling is set, at least 4 hours.