Bake a 9" pie crust (baked) and have it cooled and ready.
In a medium saucepan, scald the 3 cups whole milk .
In another large saucepan, combine the ¾ cup white sugar, 1/3 cup white flour, and ¼ tsp salt and stir with a whisk with no heat. Then add the scalded milk and whisk until well combined.
Cook the filling mixture over medium heat, stirring constantly until thickened. I use a whisk and a spatula to make sure nothing is sticking to the bottom of the pan. You do not want to scald the filling as it's thickening up.
Once it's thickened, turn the heat down to medium-low and cook for two more minutes, stirring continuously.
In a small bowl separate the 3 egg yolks and slightly beat them. Whisk in a small amount of the hot pudding mixture into the bowl (about one cup) with the egg yolks. (This is called tempering, see Holly's Helpful Hints below for more details).
When the tempered egg mixture is well combined, pour it back into the saucepan with the hot pudding mixture and stir. Cook for one minute longer.
Remove from heat and add in the 2 tbps butter and 1 tsp vanilla.
Let sit until lukewarm.
When the pudding mixture has cooled off to lukewarm, slice the 3 ripe bananas into the bottom of the pie shell. Pour the pie filling mixture into the pie and cover the bananas. Put the pie into the refrigerator.
I like to top mine with whipped cream. You can do this step when the pie is done and put in the refrigerator or you can whip the cream and add it to the top of the pie later or before serving. I piped my whipped cream onto the pie with a piping bag but you can also dollop the whipped cream onto the pie and spread it out.
Just before serving, I will add some bananas to the top and if desired, you can add some graham cracker crumbs to the top of the whipped cream.