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1
Heat 2 1/2 cups of the milk in a large saucepan over medium heat until steaming.
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2
Whisk 3/4 cup of the sugar, the cornstarch and 1/2 teaspoon salt in a large bowl.
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3
Whisk in the egg yolks and the remaining 1/2 cup milk.
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4
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.
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5
Cook over medium heat, whisking constantly, until the mixture boils.
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6
Reduce the heat to medium-low, and let the mixture come to a simmer.
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7
Continue to cook, whisking constantly, until it has thickened to a puddinglike consistency, 3 to 4 minutes.
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8
Remove from the heat and whisk in the butter and vanilla.
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9
Layer a third of the pudding, half the banana slices and half the vanilla wafers in a 9-inch square baking dish.
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10
Layer with half the remaining pudding, all the remaining banana slices and all the remaining wafers.
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11
Top with the last of the pudding.
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12
Cover with plastic wrap pressed directly on the pudding surface, and refrigerate overnight to soften the cookies.
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13
Position a rack in the upper third of the oven, and preheat to 350 degrees F. Bring the egg whites to room temperature.
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14
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.
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15
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.
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16
Spread the meringue on top of the pudding with an offset spatula so it reaches the edges of the pan.
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17
Bake until golden brown, 12 to 15 minutes.
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18
Serve warm or chilled.
