Mix the all-purpose flour, granulated sugar, and fine salt for the crust. Work in the cold butter until the mixture has pea-size pieces, then add the ice water and mix just until the dough holds together. Flatten into a disk and chill for 30 minutes.
Roll the dough into a 12-inch circle and fit it into a 9-inch pie plate. Chill for 15 minutes, then line with parchment and pie weights. Bake at 375°F for 18 minutes, remove the weights, and bake for 8 to 10 minutes more until lightly golden.
Whisk 1/2 cup of the whole milk with the cornstarch and all-purpose flour until smooth. Add the remaining milk, heavy cream, granulated sugar, and fine salt.
Cook the filling over medium heat, whisking constantly, until thick and bubbling. Continue cooking for 1 minute, then remove from the heat and whisk in the butter and vanilla extract.
Pour the hot custard into the baked crust and smooth the top. Sprinkle evenly with the ground nutmeg.
Bake the filled pie at 325°F for 15 minutes to set the surface. Cool to room temperature, then refrigerate for at least 4 hours before serving.
