Fill a large bowl halfway with ice water.
In a large saucepan over medium heat, whisk together the milk, granulated sugar, cornstarch, and salt. Bring the mixture to a simmer, whisking constantly, but do not let it boil. Simmer for 4 minutes. Remove the pan from the heat.
Whisk a ladleful of the hot milk mixture into a bowl with the lightly beaten egg yolks. Add another ladleful of milk, whisking constantly, then pour the egg mixture back into the saucepan of hot milk, still whisking constantly. Place the pan over medium-high heat and cook, whisking steadily, until the custard is thick and you see small bubbles breaking the surface, about 3 minutes.
Remove the pan from the heat, whisk in the vanilla, and place the pan in the ice bath. Stir the custard occasionally until it's fully cooled. (The custard can be made ahead and refrigerated for up to 1 day if covered tightly with plastic wrap. Make sure to press the plastic wrap against the surface of the custard to prevent a skin from forming.)
Slice the bananas—either lengthwise into elegant strips or crosswise into circles, whichever you prefer—and arrange a single layer of bananas in the pie crust. Cover them with half the pudding, then repeat the layering with the remaining banana slices and pudding.