Gather all ingredients. Preheat the oven to 350 degrees F (175 degrees C).
Combine sugar, flour, and salt in a saucepan. Gradually stir in milk.
Cook, stirring constantly, over medium heat until mixture boils and thickens. Continue to cook and stir for 2 more minutes, then remove from the heat. Place egg yolks in a medium bowl; whisk in a small amount of the hot milk mixture until smooth, then gradually whisk egg yolk mixture into the saucepan.
Cook over medium-low heat, stirring constantly until mixture coats the back of a spoon, about 2 more minutes. Remove from heat and stir in butter and vanilla.
Fill pastry shell with sliced bananas; pour pudding mixture on top to cover.
Bake in the preheated oven until filling sets, 12 to 15 minutes.
Let pie cool completely on a wire rack, then chill pie in the refrigerator for at least 1 hour before serving.
