Preheat oven to 350°F. Set a rimmed baking sheet on the middle rack to heat.
Fit the dough into a 9-inch pie plate, crimp edges, and freeze for 10 minutes.
Whisk sugar, flour, salt, and nutmeg in a large bowl.
Whisk in the beaten eggs until the mixture is smooth and glossy.
Slowly whisk in coconut milk, whole milk, and vanilla, then whisk in the melted butter.
Fold in the shredded coconut until evenly distributed.
Pour the filling into the chilled crust and place the pie on the hot baking sheet.
Bake 45 to 50 minutes, until edges are set and the center still jiggles slightly.
Shield the crust edges with foil if they brown too fast.
Cool on a rack for 2 hours. Chill 1 hour to firm before slicing.
Top with toasted coconut, slice, and serve.