In a mixing bowl, whisk together 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 half of the hot mixture into the beaten egg yolks.
Return the egg mixture to the saucepan and cook for 2 more minutes, stirring constantly.
Remove from heat and stir in butter and vanilla extract.
Arrange banana slices in the pre-baked pie crust.
Pour the pudding mixture over the bananas.
Chill in the refrigerator for at least 2 hours before serving.