In a medium saucepan, whisk together the sugar, cornstarch, and salt. Add the milk and 1 cup of the heavy cream and whisk until fully incorporated; whisk in the egg yolks. Bring the mixture to a simmer. Gently simmer (stirring constantly or the mixture will burn)until thickened, 12 to 15 minutes. Remove from heat and stir in vanilla.
Pour half of the pudding mixture into the pie crust. Slice the bananas crosswise 1/4 inch thick and arrange a layer on top of the pudding. Top with remaining pudding and smooth the top.
Place plastic wrap directly on the surface. Refrigerate for 3 hours. Beat cream until stiff. Spread over pie before serving.