Prepare the 9-inch crust according to its package or recipe directions. If blind-baking is required, prick the bottom, line the crust with parchment and pie weights, and bake at 425°F for 12–15 minutes, or until lightly golden; remove the weights and cool completely.

In a medium saucepan, whisk together the firmly packed brown sugar, flour, and salt. Gradually whisk in the whole milk so the dry ingredients dissolve without forming lumps.

Cook the mixture over medium heat, whisking frequently, until it thickens and reaches a gentle boil. Continue cooking for about 1 minute, watching carefully and avoiding prolonged boiling.

Whisk the egg yolks in a separate bowl. Slowly whisk about 1 cup of the hot milk mixture into the yolks to temper them and warm them gradually.

Pour the tempered yolk mixture back into the saucepan. Cook over medium-low heat for 1–2 minutes, stirring constantly, and do not allow the custard to boil vigorously.

Remove the saucepan from the heat and stir in the butter and vanilla until the custard is smooth. For an especially silky filling, strain it through a fine-mesh sieve.

Pour the warm filling into the cooled crust. Press plastic wrap directly onto the surface to prevent a skin from forming, then refrigerate for at least 3 hours, preferably 4 hours or overnight.

Serve the pie cold with whipped cream, adding the topping shortly before slicing. Use a sharp knife and wipe it clean between cuts for neat portions.