Make the filling base: In a medium saucepan, whisk together brown sugar, flour, and salt. Slowly whisk in the milk until smooth.
Cook: Set over medium heat, stirring constantly, until the mixture thickens and begins to bubble (about 8–10 minutes).
Temper the eggs: Slowly pour a small amount of the hot mixture into the egg yolks while whisking. Return yolks to the saucepan and cook another 2–3 minutes until thick and creamy.
Finish the pudding: Remove from heat. Stir in butter and vanilla until smooth and silky.
Assemble the pie: Pour the hot butterscotch filling into the baked pie shell. Smooth the top.
Chill: Refrigerate 3–4 hours, or until fully set.
Top & serve: Add whipped cream or meringue just before serving.