Add the flour, ¼ cup sugar, and ½ teaspoon salt to a food processor. Blend to combine, then add 1 cup of the butter (cubed and chilled) and the ice water. Pulse until the mixture forms a dough.
Take dough out of the food processor and form it into a ball. Wrap the dough in plastic wrap and chill it in the fridge for 30 minutes.
As the dough chills, preheat the oven to 425 F.
Flour a large board and roll out the dough in a circle big enough to fit into your pie pan. Place the dough in the pan, making sure that it goes fully up the sides and creates an edge.
Poke the pie crust several times with a fork (this helps it release steam while it is cooking).
Cut a round piece of parchment paper to fit in the pie and up the sides. Fill the center of this with some foil and press down into the center.
Bake crust (with the parchment paper and foil) for 15 minutes.
Remove the crust from the oven, remove the foil and parchment, and brush crust the crust with 1 beaten egg. Slide some foil underneath the pie plate and bring it up over the edges of the pie crust to protect the edges from browning too much as it continues to cook.
Reduce the oven temperature to 375 F and bake the crust for 25 to 30 minutes, or until it looks golden.
For the filling, beat 2 eggs and 2 egg whites in a medium bowl. Set aside.
Add the remaining sugar, cornstarch, remaining salt, and milk to a medium pot and cook on medium-high heat, whisking continuously, for 5 minutes.
Lower the heat to medium and cook the mixture for 5 to 10 more minutes, until the mixture has thickened.
Remove 1 cup of the milk mixture and add it to the eggs while whisking. Then add this to the pot with the rest of the milk mixture.
Transfer the custard mixture to a bowl and add the vanilla, cinnamon, and the rest of the butter. Stir until the butter is melted.
Place some plastic wrap on the surface of the mixture in the bowl and let it sit in the fridge for 1 hour.
Pour a thin layer of the chilled custard in the prepared pie crust. Thinly slice one banana and place the slices over the custard. Pour the remaining custard over the banana slices.
Cover and chill the pie in the fridge for a minimum of 4 hours, or overnight.
Right before serving, spread the whipped topping on top of the pie. Thinly slice the remaining banana and add the slices over the whipped topping. Sprinkle with additional cinnamon.