Brush the cooled pie crust with melted white chocolate and allow to set. In a medium saucepan, whisk together sugar, cornstarch, and salt. Gradually whisk in milk until smooth.

Cook over medium heat, stirring constantly, until mixture comes to a boil and thickens, about 8-10 minutes. Remove from heat.

Temper the egg yolks by whisking a small amount of hot mixture into them, then pour egg mixture back into the saucepan. Return to heat and cook for 2 more minutes, stirring constantly.

Remove from heat and stir in butter and vanilla until smooth. Pour custard into a bowl, press plastic wrap directly onto surface, and refrigerate until completely cool, about 2 hours.

Slice bananas and arrange half of them over the bottom of the prepared crust. Pour half the custard over bananas, add remaining banana slices, then top with remaining custard. Whip cream with powdered sugar until stiff peaks form and spread over pie. Refrigerate at least 2 hours before serving.