Preheat oven to 425 degrees F. Place prepared pie crust in a 9-inch pie dish. Poke the bottom with a fork or the tip of a sharp knife to help it cook thoroughly and flute the edges, if desired.
In a small bowl, combine the mascarpone cheese, ¼ cup granulated sugar, egg, and the vanilla. Beat with an electric mixer on low to medium speed until smooth. Place the bowl in the refrigerator to chill while preparing pumpkin layer.
In a medium bowl, combine the pumpkin puree, half and half, ¼ cup granulated sugar, the brown sugar, 2 eggs, and Frangelico. Stir with a wooden spoon until well combined. Add the cinnamon, ginger, salt, and nutmeg.
Pour the mascarpone cheese mixture into the crust, spreading it out evenly. Slowly pour the pumpkin mixture over the mascarpone cheese mixture. Use a knife to lightly swirl the layers together into a pretty pattern.
Bake for 15 minutes. Reduce oven temperature to 350 degrees F and bake for an additional 40 to 50 minutes, or until a knife inserted near the center comes out clean.
Remove from the oven and cool the pie on a wire rack. Cover and chill within 2 hours.
Serve with lots of whipped cream!