In a medium saucepan, whisk together 1 cup sugar, flour, and salt. In a medium bowl, whisk together milk and egg yolks. Whisk milk mixture into sugar mixture until smooth. Cook over medium heat, whisking constantly, for 10 to 12 minutes or until mixture is very thick. Remove from heat, and whisk in butter and vanilla.
In a 2½-quart baking dish, layer half of vanilla wafers, half of sliced bananas, and half of custard. Repeat layers.
Preheat oven to 350°.
In a large bowl, beat egg whites with a mixer at medium-high speed until soft peaks form. Gradually add remaining 3 tablespoons sugar, beating until stiff peaks form. Spread meringue onto custard.
Bake for 12 to 15 minutes or until golden brown. Serve warm, or cover and refrigerate for up to 2 days.