In a large mixing bowl, whisk together the sugar, cornstarch, and salt.
Add the eggs and whisk until smooth. Set aside.
In a medium pot over medium heat, bring the milk and cream to just under a boil.
Once the mixture is hot and steaming, transfer the milk and cream to a heatproof measuring cup. Very slowly pour it into the bowl from step two while whisking constantly and vigorously. This gradual process tempers the eggs and ensures a smooth, creamy pudding.
When all of the liquid has been added, pour the mixture back into the pot. Place it over low heat and whisk continuously until the pudding thickens and just begins to bubble.
As soon as it reaches a boil, whisk vigorously for 30 to 60 seconds.
Immediately strain the vanilla pudding through a fine-mesh sieve into a clean mixing bowl to remove any lumps or bits of cooked egg.
Whisk in the butter and vanilla extract until smooth.
Press a piece of plastic wrap directly against the surface to prevent a skin from forming.
Chill the pudding for at least 1 hour. This is so that the pudding is cool enough it doesn’t begin cooking the banana slices.
Once the vanilla pudding has chilled to at least room temperature, fold in the banana slices.
Pour the vanilla cream and banana mixture into your prepared, fully-baked pie crust. Make sure all the banana slices are fully covered by the vanilla cream so that they don’t turn brown before serving.
Chill the pie for at least 3 hours to allow the filling to fully cool and set.