Line bottom of chilled crust with a layer of banana slices and set aside.
Place milk, sugar, 3 tablespoons cornstarch, and salt into a saucepan and place over medium heat. Whisk mixture until smooth and bring to a simmer, occasionally stirring with a wooden spoon until thick enough to coat the back of the spoon, 5 to 6 minutes.
In a small bowl, whisk together remaining tablespoon cornstarch and egg yolks until mixture is smooth.
Remove 1 cup hot milk from the pot and whisk into the yolks (in a mixing bowl) in a very slow and steady drizzle, making sure not to scramble the yolks.
Lower heat to medium-low and pour tempered yolk mixture back into the pot and stir together until smooth.
Cook the mixture for 2 to 3 more minutes, before removing from the heat and stirring in the butter and vanilla. Stir until butter melts and is evenly incorporated into the pudding. Mixture should be very thick.
Place pot of pudding directly into an ice bath (to stop the cooking process), stirring occasionally until slightly cooled, 10 to 25 minutes.
Pour pudding into prepared crust and spread into an even layer and refrigerate between 2 hour and 6 hours (until filling has set and pie is cold) and up to 24hours.