In a medium saucepan, combine sugar, cornstarch and salt. Gradually add milk while stirring. Cook over medium heat, stirring constantly until mixture begins to thicken and bottom of pan can be briefly glimpsed while stirring. Remove from heat.
Temper yolks by adding a small bit of hot milk mixture to yolks. Stir. Add a bit more. Stir again. Pour yolk mixture into hot milk mixture. Stir until well incorporated.
Return to medium heat. Cook an additional 2 minutes, stirring constantly. Reduce heat as needed to maintain a low simmer.
Remove from heat. Add butter and vanilla. Stir until smooth. Set aside.
Slice bananas into baked pie crust. (About 1/2" thick).
Stir pudding well. Pour over bananas, being sure to cover bananas completely. Cover with plastic wrap immediately to prevent pudding skin from forming.
Cool in refrigerator 2 hours, or until cold.
Cover pie with whipped topping immediately before serving.
Cover leftovers tightly with plastic wrap and store in the fridge up to 2-3 days.
For a creamy consistency of vanilla pudding, use 1/4 cup cornstarch as in the recipe.
For a thicker, more custard-like consistency of vanilla pudding, increase cornstarch to 1/3 cup.
If substituting with homemade whipped cream, use 3 cups of sweet whipped cream, made by beating 1 1/2 cups heavy whipping cream, 1/2 cup powdered sugar and 1 1/2 teaspoons vanilla extract to stiff peaks.