Preheat the oven to 350 °F.
Lightly beat the egg yolks. Combine 3⁄4 cup of the sugar and the flour in a medium pot. Gradually stir in the milk, followed by the egg yolks and salt. Cook over medium heat 15 to 20 minutes, stirring constantly, until thickened. Remove the pot from the heat and add the butter and 1 teaspoon of the vanilla. Set aside to cool slightly. Arrange half of the wafers on the bottom of a 2-quart baking dish. Top with half of the banana slices. Spoon half of the pudding mixture over the bananas. Repeat with remaining wafers, bananas, and pudding.
In a clean bowl, beat the egg whites with the remaining 1⁄4 cup of sugar and 1 teaspoon of vanilla until they form stiff peaks. Spread with a rubber spatula over the pudding, sealing the edges. Bake until the topping is light golden, 15 to 20 minutes. Remove from the oven and let cool. Serve cooled.
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