Pour water into a medium saucepan to come 1½" up the sides and bring to a bare simmer over medium-low heat. Whisk eggs, brown sugar, and salt in a large heatproof bowl just to combine (or use the bowl of a stand mixer if you’ve got one). Set over saucepan (bowl should not touch water) and heat egg mixture, stirring constantly with a heatproof rubber spatula, until sugar dissolves, about 4 minutes (mixture will no longer feel gritty when rubbed between your fingers). If you are concerned about undercooked eggs, heat mixture longer, still stirring constantly, until an instant-read thermometer registers 165°, about 3 minutes more. Remove bowl from heat and add vanilla to egg mixture. Using an electric mixer, beat on high speed until thick and pale and doubled in volume, about 10 minutes. (Or, if using a stand mixer, beat with the whisk attachment, 6–8 minutes.)
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