Bake the pie crust according to package instructions. Cool while you prepare the filling.
1 9-inch pie crust
Add the egg and egg yolks to a medium bowl and whisk well. Set aside
1 large egg, 3 egg yolks
In a large saucepan, whisk the sugar, flour, and salt together until combined. Whisk in the milk and cook over medium heat, stirring frequently, until it begins to bubble and thicken.
½ cup granulated sugar, 3 tablespoons all-purpose flour, ½ teaspoon kosher salt, 2½ cups whole milk
Remove the saucepan from the heat and pour ½ cup of the milk mixture into the eggs, whisking constantly, to temper. Whisk the egg mixture back in the saucepan and return it to medium-low heat, bringing it to a gentle boil. Cook over low heat for 2-3 minutes, whisking frequently, then remove from stovetop and stir in butter and vanilla. Press a piece of plastic wrap on the surface and allow to cool for 10-15 minutes.
2 tablespoons unsalted butter, 1 teaspoon pure vanilla extract
Line the bottom of the baked pie crust with half of the sliced bananas. Pour in half of the pudding mixture, then layer the remaining sliced bananas. Top with the rest of the pudding. Place a piece of plastic wrap directly on top of the filling and refrigerate for 4-24 hours.
3 bananas
When ready to serve, spread or pipe whipped cream (homemade or store-bought) on top of the pie, then garnish with more banana slices.
2 cups whipped topping, 1 banana