Whisk 1/4 cup of the sugar and the cornstarch together in a medium saucepan. Whisk in 1/4 cup of the milk and the egg yolks until smooth. Whisk in the remaining 2 cups milk and bring to a boil over medium-high heat, stirring and scraping the bottom and corners of the pan often with a heatproof rubber spatula. Once boiling, stir constantly, until thick, about 2 minutes. Remove from the heat and whisk in 1 tablespoon of butter and the vanilla. Divide the pudding among 4 small glasses.
Combine the remaining 1/2 cup sugar with 2 tablespoons water in a medium skillet and stir to moisten all of the sugar. Cook over medium-high heat, swirling the skillet so the sugar cooks evenly, until dark golden brown, 5 to 7 minutes. Add the remaining 3 tablespoons butter, the bananas, and salt and cook, stirring, until the bananas are slightly softened, about 2 minutes.
Spoon the bananas on top of the pudding and serve warm.
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