Heat oven to 375°F and place a rack in the lower-middle position.
Fit the pie dough into a 9-inch pie plate, trim and crimp edges, and dock the bottom with a fork.
Chill the lined pie shell for 10 minutes to help prevent shrinkage.
Line the crust with parchment, fill with pie weights, and bake 12 minutes. Remove weights and bake 3 minutes more until just dry.
Whisk granulated sugar, brown sugar, cornstarch, and salt in a medium saucepan.
Slowly whisk in the heavy cream until the mixture is smooth with no dry spots.
Cook over medium heat, whisking constantly, until thickened and bubbling, 6–8 minutes.
Take off the heat and whisk in the butter and vanilla until glossy.
Pour the hot filling into the warm crust and smooth the surface with a spatula.
Dust the top evenly with cinnamon and nutmeg.
Bake 15–18 minutes until the edges bubble slightly and the center has a gentle jiggle.
Cool on a rack until room temperature, then chill at least 2 hours to fully set before slicing.