In a medium saucepan, combine sugar, flour, and salt, whisking them together until well-mixed. Gradually whisk in 3 cups of milk over medium heat, stirring consistently to prevent lumps. Cook this mixture for about 10 minutes, or until it thickens and begins to bubble, resembling a thick custard.

While your pudding mixture is thickening, crack and beat 3 large eggs in a separate bowl. Once the pudding is thickened, slowly temper the eggs by whisking in a spoonful of the hot pudding. Carefully mix the tempered egg mixture back into the saucepan, stirring continuously. Return the mixture to medium heat and cook for an additional 2 minutes until it just starts to gently boil, stirring frequently.

After reaching a gentle boil, remove the saucepan from the heat. Stir in 1 tablespoon of pure vanilla extract, mixing well to infuse the flavor throughout the pudding. Allow the pudding to cool for about 15 minutes while stirring occasionally.

Grab an ungreased 8-inch square baking dish. Start layering with half of the vanilla wafers, followed by half of the banana slices. Then, pour half of the cooled pudding mixture over the layers. Repeat the layers with the remaining ingredients.

Cover the banana pudding with plastic wrap and refrigerate it for at least 4 hours, but overnight is best for richer flavors.

Before serving, take the remaining vanilla wafers and crush them into coarse crumbs. Sprinkle these crushed cookies on top of your chilled banana pudding.