Preheat the oven to the temperature specified on your pie crust package.

Combine sugar, flour, and salt in a medium saucepan. Gradually whisk in the whole milk until smooth.

Heat the mixture over medium heat, stirring continually until it thickens and bubbles, about 5-7 minutes.

Temper the egg yolks by gradually adding a few spoonfuls of the warm pudding mixture to the beaten yolks, whisking constantly.

Return the tempered yolks to the saucepan and cook until thickened, around another 2-3 minutes. Stir in butter and vanilla extract until smooth.

Layer sliced bananas in the cooled pie crust. Pour half of the pudding mixture over the bananas, then add another layer of bananas followed by the remaining pudding.

Chill the pie in the refrigerator for at least 5 hours or overnight.

Serve slices of pie topped with Cool Whip and additional banana slices.