Place a rack in the lower-middle position in the oven and preheat to 425 F.
Roll out pie crust, transfer to a 9 inch pie pan, and dress the edges. Chill for 30 minutes while the oven heats.
Dock the dough with a fork, then bake until the edges just start to brown, about 15 minutes. Remove from oven and set aside. Keep oven on at 425 F.
Combine the pumpkin purée, sugars, salt, and spices in a food processor or blender. Blend until brown sugar is completely incorporated.
Lightly cook purée over medium-high heat, stirring frequently, until mixture is very steamy and too thick to stop stirring without making a mess, 5 minutes or less. Remove pan from heat.
In a separate bowl, whisk together the milk, cream, and eggs. Add a spoonful of the hot pumpkin mixture to the milk mixture and stir. This will temper the eggs.
Slowly whisk the milk mixture into the purée. Whisk in the vanilla.
Fill the pie crust with the pumpkin mix.
Transfer pie to oven, then turn heat down to 375 F. Bake until pie is puffed in the middle, but still jiggles slightly, about 50-60 minutes. If the crust begins to darken before the middle is cooked, cover the crust with foil.
Let the pie cool completely before serving, at least 4 hours but up to over night.
When ready to serve, prepare the whipped cream by whisking together the heavy cream and sugar. It can be done by hand, but is much easier with an electric or stand mixer.