In a double boiler, whisk together the brown sugar, cornstarch, salt, and milk. Stir constantly over medium heat for about 20 minutes until the mixture thickens.
Slowly whisk in the beaten egg yolks, continuing to stir until the mixture resembles a thick custard.
Preheat the oven to 400°F (200°C).
Remove the custard mixture from the heat, then stir in butter and vanilla extract until smooth.
Pour the filling into the pre-baked pie crust, spreading evenly.
Bake for 5 minutes, or until the top of the pie begins to brown slightly.
Allow the pie to cool completely before slicing and serving.