Preheat the oven to 225 degrees F. Line baking trays with parchment paper or nonstick silicone mats.
Place the egg whites and sugar in a large metal electric-mixer bowl set over simmering water. Whisk constantly until the sugar melts and the mixture is very thin and warm. Remove the bowl from the heat and whisk it with an electric mixer on high speed until stiff peaks form, about 5 minutes. Add the vanilla extract.
Drop the meringue into mounds on the prepared tray with a tablespoon, or fill a disposable piping bag with the meringue and pipe through a star tip.
Bake until firm, about 1 hour. Turn off the oven and let the drops continue drying until cooked, about 1 more hour. For a chewier texture, shorten the baking time.