Preheat the oven to 375 degrees F (190 degrees C).

Make the pudding: Combine sugar, flour, and salt in a bowl. Set egg whites aside for meringue.

Beat egg yolks in a heavy saucepan over medium heat. Add sugar mixture alternately with milk and vanilla, stirring constantly. Bring to a simmer and cook until slightly thickened, 8 to 10 minutes. Add butter; continue stirring until thickened to a pudding-like consistency, 5 to 7 minutes more.

Line the bottom of a baking dish with 1/2 of the wafers. Layer 1/2 of the banana slices over wafers and cover with 1/2 of the pudding. Repeat layers once more.

Make the meringue: Beat reserved egg whites in a glass, metal, or ceramic bowl, gradually adding cream of tartar, until soft peaks form. Add sugar and continue beating on high speed until stiff peaks form. Fold in vanilla. Spoon meringue over pudding.

Bake in the preheated oven until meringue browns, 15 to 20 minutes. Let cool before serving.