Step 1: In a medium saucepan, melt the butter over medium heat. Stir in the brown sugar, flour, and salt. Cook for 2-3 minutes, stirring constantly, until the mixture is well combined and slightly bubbly.
Step 2: In a separate bowl, whisk the egg yolks with about 1/4 cup of the milk. Gradually whisk in the remaining milk until well combined.
Step 3: Slowly pour the milk and egg mixture into the saucepan with the butterscotch mixture, whisking constantly to prevent the eggs from scrambling. Continue to cook over medium heat, whisking constantly, until the mixture thickens and comes to a boil, about 8-10 minutes.
Step 4: Once thickened, remove the saucepan from the heat and stir in the vanilla extract.
Step 5: Pour the hot butterscotch filling into the pre-baked pie crust. Smooth the top with a spoon or spatula.
Step 6: Cover the pie loosely with plastic wrap, pressing it directly onto the surface of the filling to prevent a skin from forming. Chill in the refrigerator for at least 3-4 hours, or until completely set.
Step 7: Once chilled, remove the plastic wrap. If desired, top with whipped cream or a meringue before serving.