In a saucepan, combine brown sugar, cornstarch, and salt.
Gradually whisk in milk until smooth.
Cook over medium heat, stirring constantly, until mixture thickens and boils.
Boil for 1 minute, then remove from heat.
Gradually whisk a small amount of hot mixture into egg yolks, then return all to the pan, whisking constantly.
Bring to a gentle boil; cook and stir for 2 minutes.
Remove from heat; stir in butter and vanilla until smooth.
Pour filling into pre-baked pie crust.
Chill for at least 4 hours before serving.