Prepare the Custard

In a medium saucepan, whisk together the sugar, cornstarch, and salt.

Gradually whisk in the milk, making sure there are no lumps.

Cook over medium heat, stirring constantly, until the mixture starts to thicken and bubble—about 6–8 minutes.

Temper the Eggs

In a small bowl, beat the egg yolks.

Slowly pour ½ cup of the hot custard mixture into the yolks, whisking constantly to temper them (this prevents the eggs from scrambling).

Pour the tempered yolks back into the saucepan with the rest of the custard and stir continuously for another 2–3 minutes, until thickened further.

Finish the Custard

Remove the pan from the heat. Stir in the butter and vanilla extract until smooth and fully incorporated.

Let the custard cool slightly while you prepare the crust and bananas.

Assemble the Pie

Slice the bananas and evenly layer them over the bottom of the pre-baked pie crust.

Pour the warm custard over the banana slices and smooth out the top with a spatula.

Chill

Cover the surface of the custard directly with plastic wrap to prevent a skin from forming.

Refrigerate the pie for at least 4 hours, or until fully chilled and set.

Add Whipped Cream

Before serving, top the pie with whipped cream, either spread over the top or piped decoratively.

Slice and serve chilled.