Separate the yolks from the whites of 3 of the eggs; set aside the whites. Add the remaining whole egg to the yolks.
In a saucepan, whisk together 1/2 cup sugar, the flour and 1/2 teaspoon salt. Stir in the whole egg and 3 yolks, and then stir in the milk. Cook uncovered, stirring often, until the mixture thickens, about 10 minutes. Remove from the heat and stir in the vanilla.
Preheat the oven to 425 degrees F.
Spread a thin layer of the pudding in a 1 1/2-quart casserole dish. Arrange a layer of vanilla wafers on top of the pudding. Thinly slice the bananas crosswise, about 1/8 inch thick, and arrange a layer of banana slices over the wafers. Spread one-third of the remaining pudding over the bananas and continue layering wafers, bananas and pudding, ending with pudding.
To make the meringue, beat the reserved egg whites with a pinch of salt until they are stiff. Gradually beat in the remaining 1/4 cup sugar and continue beating until the whites will not slide out of the mixing bowl when it is tilted.
Spread the meringue over the pudding with a spatula, making a few decorative peaks on top, and bake until the meringue is lightly browned, 5 minutes.