Make the Pudding: In a medium bowl, whisk together the sugar, flour, and salt. Whisk in ½ cup of the milk and the egg yolks. In a large saucepan, place the remaining 3 cups milk. Bring to a simmer over medium heat, stirring frequently. Whisk 1 cup of the hot milk into the egg mixture. Whisk the egg mixture into the remaining hot milk in the saucepan. Bring to a boil over medium heat, stirring constantly. Cook, stirring constantly, until thickened, about 3 minutes more. Remove from heat. Add the butter and vanilla, stirring until the butter is melted. Set aside to cool slightly while you prepare the remaining ingredients.
Assemble the Dish: Arrange vanilla wafers in the bottom of a 9x13-inch baking dish (you can reserve some to garnish, if you wish), and top with the banana slices. Pour the pudding through a fine-mesh sieve over the bananas and spread into an even layer.
Make the Meringue: In a large bowl, beat the egg whites and cream of tartar with a mixer on high speed until soft peaks form. Gradually add the sugar, beating until stiff peaks form. Spoon the meringue over the pudding, spreading it right up to the edges of the pan.
Bake until the meringue is lightly browned, 10 to 15 minutes. Garnish with vanilla wafers, if desired. Let cool for 10 minutes before serving.
I don't recommend using super ripe bananas in this recipe, as they will break down too much when the pudding is in the oven.