In a large saucepan, whisk together the sugar and cornstarch until well combined. Set aside.
In a separate large mixing bowl, whisk together the half and half and egg yolks until fully combined. Slowly pour the wet ingredients into the saucepan with the sugar and cornstarch mixture and whisk until fully combined.
Place the saucepan over medium heat and bring to a boil stirring continuously (I suggest using a rubber spatula to scrape around the bottom and sides of the pan a few times). Allow to boil for one minute, the mixture should be pretty thick at this point. Remove from the heat and stir in one tablespoon of butter at a time, then stir in the vanilla extract.
Strain the mixture through a fine-mesh strainer into a heat-safe bowl. Cover with plastic wrap, placing the plastic wrap directly on the surface of the custard, and allow it to stand for 30 minutes.
Remove the plastic wrap, stir the filling, scoop half of the filling into the cooled pie crust, and spread it out into one even layer. Add the banana slices in one even layer on top of the filling, then scoop the rest of the filling on top of the sliced bananas and spread it out into an even layer. Cover tightly with plastic wrap and refrigerate the pie for 3 hours or until completely chilled and firm.