Prepare the Pudding: In a medium saucepan, combine the sugar, flour, and salt. Gradually whisk in the milk until smooth. Cook the mixture over medium heat, stirring constantly, until it thickens and begins to bubble. Reduce the heat to low and continue cooking for an additional 2 minutes.
Temper the Eggs: Remove the saucepan from heat. In a small bowl, whisk the eggs together. Gradually add a small amount of the hot pudding mixture to the eggs, whisking constantly to prevent the eggs from scrambling.
Finish the Pudding: Return the egg mixture to the saucepan with the remaining hot pudding mixture. Cook over medium-low heat, stirring constantly, until the pudding starts to gently boil. Continue cooking for about 2 minutes. Stir in the vanilla extract, then remove the pudding from the heat. Allow the pudding to cool for about 15 minutes, stirring every few minutes to prevent a skin from forming.
Assemble the Pudding: In an ungreased 8-inch square pan, layer about 25 vanilla wafers on the bottom. Top the wafers with half of the banana slices. Pour half of the pudding mixture over the bananas. Repeat the layers with the remaining wafers, bananas, and pudding, finishing with a layer of pudding on top.
Chill the Pudding: Cover the dish with plastic wrap and place it in the refrigerator. Allow the pudding to chill for at least 4 hours, but preferably overnight, to set the flavors.
Serve: Before serving, crush the remaining vanilla wafers and sprinkle them over the top of the pudding. Serve cold and enjoy this classic homemade dessert!