Preheat oven to 425 degrees and place rack in the middle.
Roll out pie dough to fit into a 9" pie dish and pinch the edges of the crust like shown above. Chill crust in the refrigerator or freezer for at least 15 minutes.
After crust has been refrigerated or placed in freezer, line inside of pie with parchment paper and fill with pie weights or dried beans to prevent shrinking. Bake for 7-8 minutes. Remove from the oven, remove parchment paper, and prick with a fork. Return to oven and bake for 10-12 minutes or until golden brown. Let cool.
For the filling: In a large saucepan, add the sugar, flour, and salt. Gradually stir in the milk and cook over medium heat until mixture is boiling and thickens. Cook for 2 minutes.
Remove from heat. Stir a small amount of the milk mixture into a separate bowl with egg yolks to temper them (about 2 tablespoons). This brings the temperature of the eggs up so that when they are added into the pan they don't scramble. Add the tempered eggs back into the milk mixture and cook for 2 minutes stirring constantly. Whisk constantly until the texture is thick. This is important! If the mixture is not cooked until thick it won't set up properly.
Remove mixture from heat and add the butter and vanilla.
Transfer to a separate bowl. Allow mixture to cool to room temperature.
Slice bananas and place them in the pie shell and add the filling on top.
For the whipped cream: Place cream and sugar in a chilled mixing bowl and mix on high for 1-2 minutes. Add the vanilla and beat for an additional 1-2 minutes or until soft peaks have formed. Don't whip too much or you will end up will cottage cheese looking whipped cream.
Spread the whipped cream over the top of the filling and chill until ready to serve.
Notes
CAN I MAKE BANANA CREAM PIE AHEAD OF TIME?
You can make this Banana Cream Pie ahead of time up to step 10. You can make the crust and the filling up to two days ahead of time and store separately. Because the bananas will start to brown you don't want to slice and assemble them until the day of serving.