In a medium saucepan, whisk together butter, flour, sugar, and salt.
Gradually whisk in milk. Cook over medium heat, stirring constantly, until the mixture thickens and comes to a boil. Boil for 1 minute.
In a small bowl, whisk the egg yolks. Temper the egg yolks by slowly whisking about 1 cup of the hot milk mixture into the yolks. Then, pour the tempered yolks back into the saucepan.
Cook for another 1 to 2 minutes, stirring constantly, until the pudding is thick. Do not boil.
Remove from heat and stir in vanilla extract.
Layer half of the cookies in the bottom of a 9x13 inch baking dish. Top with half of the sliced bananas. Pour half of the pudding mixture over the bananas.
Repeat the layers with the remaining cookies, bananas, and pudding.
Cover with plastic wrap, pressing it directly onto the surface of the pudding to prevent a skin from forming. Chill for at least 4 hours, or preferably overnight.
Notes
For an extra special touch, you can top with whipped cream before serving.