Heat oven to 375°F. Fit dough into a 9-inch pie plate and crimp edges.
Line crust with foil and fill with pie weights. Bake 10 minutes.
Remove foil and weights. Bake 3 minutes more. Reduce oven to 350°F.
Whisk granulated sugar, brown sugar, flour, salt, and cinnamon in a saucepan.
Slowly whisk in heavy cream and milk until smooth and lump-free.
Cook over medium heat, whisking often, until thick and bubbling, 6–8 minutes.
Remove from heat. Whisk in butter and vanilla until silky and glossy.
Pour filling into warm crust. Smooth the top and dust lightly with nutmeg.
Bake at 350°F until edges are set and center jiggles slightly, 22–28 minutes.
Cool 1 hour on a rack. Chill at least 2 hours before slicing.