Preheat your oven to 350°F (175°C). Bake the pie crust according to package instructions or your recipe if homemade. Let it cool.
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 and comes to a boil. Boil for 1 minute, then remove from heat.
Stir in the butter until melted. Gradually whisk a small amount of the hot mixture into the beaten egg yolks, then return all to the saucepan, whisking constantly.
Bring to a gentle boil and cook for 2 minutes, stirring constantly. Remove from heat and stir in vanilla extract.
Pour the filling into the baked pie crust. Cool to room temperature, then refrigerate for at least 2 hours before serving.