Start by preparing your baked 9-inch pie crust, setting it aside for later use.

In a large saucepan, warm the milk until it begins to steam, just before it reaches a boil. Meanwhile, in another saucepan, mix together the sugar, flour, and salt. Gradually add the hot milk to this dry mixture, stirring constantly to ensure a smooth consistency.

Place the saucepan over medium heat and continue to stir until the mixture thickens. Once it reaches the desired consistency, cover the pan, and cook for an additional two minutes, stirring occasionally to prevent any sticking.

In a small bowl, have your slightly beaten egg yolks ready. Slowly introduce a small portion of the hot mixture into the yolks, blending thoroughly to temper them. Once combined, pour the yolk mixture back into the saucepan with the rest of the hot filling, stirring continuously. Cook for another minute until fully integrated, then remove from the heat.

Add the butter and vanilla extract, stirring until the butter has melted and the mixture is smooth. Allow the filling to cool to lukewarm temperature.

When ready to assemble, slice the bananas and arrange them evenly in the bottom of the baked pie crust. Pour the warm filling over the banana slices, ensuring they are fully covered.

If you prefer, you can top the pie with a meringue using the leftover egg whites, or simply allow the pie to cool completely before serving.