Heat milk in top of a double boiler set over simmering water. In a small bowl, whisk together 3⁄4 cup of the sugar, cornstarch, egg yolks, and enough water to make a soupy paste (about 1⁄4 cup). Whisk egg yolk mixture into milk. Cook, stirring constantly, until thickened to the consistency of pudding, 30–40 minutes. Remove from heat; stir in butter and vanilla. Let cool slightly.
Preheat the oven to 325°. Cover bottom of a 9" × 13" baking dish with half the vanilla wafers. Slice the bananas into 1⁄2"-thick rounds and spread half of them over wafers. Cover bananas with half the pudding. Spread remaining wafers over pudding, followed by remaining bananas. Pour rest of the pudding over top.
Place egg whites and cream of tartar in a chilled metal bowl. Using an electric mixer on slow to medium speed, beat until soft peaks form. Add remaining 3 tbsp. sugar 1 tbsp. at a time and continue to beat until stiff but not dry peaks form to make a meringue. Spread meringue over pudding and bake until lightly browned, 20–30 minutes. Serve warm, at room temperature, or chilled.