In a small bowl, mix the cornstarch with ½ cup milk.
In a large saucepan, mix the remaining milk with the half-and-half, sugar and salt. Heat over medium heat, stirring occasionally, until the mixture comes to a boil. Gradually pour about ¼ of the hot mixture into the eggs and whisk well. Pour the egg mixture into the saucepan along with the cornstarch mixture, lower the heat to medium-low, and bring to a boil, stirring constantly, until the mixture thickens to the consistency of pudding, about 1 minute. Take the mixture off the heat and stir in the butter and vanilla. Push the mixture through a fine sieve into a clean bowl.
Peel the bananas and slice them diagonally in about ¼ inch slices. Reserve a few slices for garnish and layer the rest into the bottom of the cooled pie shell. Pour the pudding over the top, smooth with a spatula, let cool, and then refrigerate until cold and firm, about 4 hours.
Make the topping:
Whip the cream with the sugar and vanilla until stiff peaks form. Spread over the top of the pie, garnish with banana slices, and serve.