Using the back of a fork, mash one of the bananas very well in a medium saucepan. Add the eggs, egg yolks and sugar to the saucepan (on the countertop, NOT heated on the stove yet). Using a hand mixer on a medium-high speed, beat the mixture until pale in color and frothy.
In a separate bowl, make a slurry by combining the cornstarch with just enough milk to make it dissolve without and no lumps remain. Then add the remaining milk and vanilla, stirring very well.
Place the saucepan on the stove over a low heat and add the milk mixture to the egg and banana mixture while stirring continuously. Gradually turn the heat up to medium over the course of several minutes while continuing to stir and scrape the bottom with a spatula. At about the 10 minute mark, you should start to notice it becoming more fragrant and starting to thicken.
When it looks like the consistency of pudding, remove it from the heat and set it aside for a moment while you slice a banana and place the pieces in a single layer in the bottom of the prepared crust.
Pour the filling mixture on top of the layer of bananas in the prepared crust. Cover with plastic wrap to prevent a skin from forming on the pie filling and return it to the fridge to chill through. This will take about 3 hours.