Heat 2 1/2 cups of the milk in a large saucepan over medium heat until steaming.
Whisk 3/4 cup of the sugar, the cornstarch and 1/2 teaspoon salt in a large bowl. Whisk in the egg yolks and the remaining 1/2 cup milk.
Whisk half of the hot milk into the egg mixture until smooth, and then gradually whisk the egg-milk mixture into the milk in the saucepan. Cook over medium heat, whisking constantly, until the mixture boils. Reduce the heat to medium-low, and let the mixture come to a simmer. Continue to cook, whisking constantly, until it has thickened to a puddinglike consistency, 3 to 4 minutes. Remove from the heat and whisk in the butter and vanilla.
Layer a third of the pudding, half the banana slices and half the vanilla wafers in a 9-inch square baking dish. Layer with half the remaining pudding, all the remaining banana slices and all the remaining wafers. Top with the last of the pudding. Cover with plastic wrap pressed directly on the pudding surface, and refrigerate overnight to soften the cookies.
Position a rack in the upper third of the oven, and preheat to 350 degrees F. Bring the egg whites to room temperature. Beat the egg whites and cream of tartar in the bowl of a stand mixer fitted with a whisk attachment on medium speed until frothy. Increase the speed to medium-high, and slowly sprinkle in the remaining 1/2 cup sugar a tablespoon at a time, whipping for 15 seconds after each addition, until glossy firm peaks form. Spread the meringue on top of the pudding with an offset spatula so it reaches the edges of the pan. Bake until golden brown, 12 to 15 minutes. Serve warm or chilled.
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