Preheat your oven to 350°F (175°C).
In a saucepan, combine brown sugar, cornstarch, and salt.
Gradually whisk in the milk until smooth.
Cook over medium heat, stirring constantly until the mixture thickens.
In a separate bowl, whisk the egg yolks.
Slowly add a bit of the hot mixture to the egg yolks to temper them.
Pour the egg yolk mixture back into the saucepan.
Cook for an additional 2-3 minutes while stirring.
Remove from heat and stir in butter and vanilla extract.
Pour the filling into the pre-made pie crust and refrigerate until set.