Line a 9-inch pie pan with crust and then crimp the edges decoratively. Line and fully bake the crust (see Notes).
To make the filling:
In a medium saucepan, combine the sugar, flour, and salt. Use a fork or a whisk to mix them together. Add the milk to the saucepan, and cook over medium heat, stirring often, until you have a thickened, smooth sauce, about 10 minutes.
Place the egg yolks in a small bowl. Add about 1/4 cup of the hot milk mixture to the egg yolks, stirring constantly as you pour the milk slowly into the bowl. Mix well, Pour the warmed egg yolks into the saucepan of sugar and milk, stirring constantly to help them mix in smoothly.
Cook, stirring often, 2 to 3 minutes more, until the filling is thick and smooth. Remove it from the heat and stir in the butter and vanilla, mixing everything evenly and well. Pour the filling into the baked piecrust and set aside.
To make the meringue:
Heat the oven to 350 degrees F. Beat the egg whites in a medium bowl with an electric mixer on medium speed until foamy. Increase the speed to high, and add the sugar gradually, about 1 tablespoon at a time. Beat the egg whites until the meringue is thick, shiny, and able to hold curly firm peaks. Scoop the meringue onto the filling. Spread it out to seal the edges to the crust, mounding it up in the middle. Use the back of a spoon to swirl it up into curly shapes.
Place the pie on the middle rack of the oven. Bake until the meringue has turned a beautiful golden brown, 10 to 15 minutes. Place the pie on a cooling rack or a folded kitchen towel and let it cool. Serve at room temperature, or refrigerate and serve chilled.