Preheat the oven to 350 degrees F. Spray a 9-inch-diameter glass pie dish with nonstick spray to coat lightly.
Unroll the pie crust and press it gently into the pie pan creating a high standing edge on the rim of the dish. Crimp the edge decoratively. Pierce the crust evenly with the tines of a fork to prevent puffing while baking.
Bake the crust until the edge is browned and the center of the crust is pale golden, about 25 minutes. Transfer the crust to a work surface. Brush the sides and bottom (but not the crimped edge) with the apricot preserves to coat.
In a food processor, combine the persimmon puree, pumpkin, sugar, mascarpone, cream, cornstarch, cinnamon and salt and blend well. Add the eggs and blend for 5 seconds. Pour the filling into the prepared crust. Place the pie into the oven and, if the edge will over-brown, stand a foil collar directly on the oven rack and wrap it around the dish to protect the crust.
Bake the pie until slits or cracks appear around the edge and the center is set, about 35 minutes. Cool the pie on a rack. (Can be made 1 day ahead. Refrigerate until cold, then tent loosely with foil.)
Cut the pie into wedges and serve with whipped mascarpone if desired.