1. In a heavy saucepan, whisk together the sugar, cornstarch, and salt. Gradually whisk in the milk, then whisk in the egg yolks.
2. Bring the mixture just barely to a simmer while stirring constantly with a wooden spoon.
3. Remove from the heat and whisk until smooth, scraping the corners of the saucepan well.
4. Return to the heat and bring back to a simmer. Cook for one minute while whisking constantly, then remove from the heat and whisk in the butter and the vanilla. Allow to cool slightly (the filling does not need to be cool to fill the pie; in fact it is better if you do not allow it to cool too much).
5. Thinly slice 2-3 ripe (but not overripe) bananas. Spoon 1/3 of the filling into the crust and place 1/3 of the bananas on top. Add another 1/3 of the filling and then more bananas. Finish with the last of the filling and the rest of the bananas. You can top with homemade (or store-bought) whipped cream, if you like.