Line the bottom of a 9" x 9" baking dish with a layer of vanilla wafers.
Peel the bananas and slice into 3/8-inch rounds. Cover the banana slices with plastic wrap to keep them from darkening and quickly make your pudding.
Combine the sugar, flour and salt in a bowl, and stir well to mix. Mash out any flour lumps with the back of your spoon. Set aside.
In a heavy saucepan, beat the egg yolks well (just use a fork or whisk, but beat them well). Over medium heat, add the flour mixture to the egg yolks, alternately with the milk and vanilla, stirring constantly. Bring to a gentle boil and, when the mixture begins to thicken, add the butter, continuing to stir. Keep boiling and stirring until mixture reaches a nice pudding consistency. [Note: If you're working with an electric cooktop, adjust the heat so that it's hot enough to boil, but not so hot that the pudding scorches.] Remove from heat.
Place a layer of banana slices in the baking dish on top of the vanilla wafers. Don't stint and put one slice of banana per wafer. Line those banana slices up edge-to-edge. Pour, spreading as necessary, half of the pudding over the banana layer. Put down another layer of vanilla wafers, another layer of banana slices, and cover with the remaining pudding.
Beat the egg whites at high speed until they form soft peaks. Add the cream of tartar. At high speed, gradually add the sugar, a tablespoon at a time, and beat until stiff peaks form. Fold the vanilla into the meringue, and spread the meringue over the pudding, sealing it at the sides of the dish.
Bake in a preheated 375°F oven until meringue browns, 12 to 15 minutes, depending upon your oven.