Preheat the oven to 225°. Soak the tapioca in 1/2 cup of milk for 1 hour. Lightly butter four 6-ounce ramekins.
In a small saucepan, combine the mango with the water, vanilla bean and 1 tablespoon of the sugar. Simmer over low heat, stirring occasionally, until the mango softens, about 5 minutes. Remove the pan from the heat. Remove the vanilla bean from the saucepan and stir in the lime juice and zest.
In a medium saucepan, combine the tapioca and any of its soaking milk with the remaining 1 cup of milk. Simmer over low heat, stirring occasionally, until the tapioca is tender but still chewy, 30 to 35 minutes.
In a small saucepan, whisk the coconut milk with 1/3 cup of the sugar and the salt and bring to a boil over moderate heat. Transfer to a medium bowl. Stir in the tapioca and chill in the refrigerator or quick-chill by setting the bowl in a larger bowl of ice water.
In a stainless-steel bowl, whisk the egg whites with the remaining 2/3 cup of sugar. Set the bowl over but not in a medium pan of simmering water and whisk constantly until the whites are slightly warmer than body temperature, about 100° on a candy thermometer. Remove the bowl from the water and beat the egg whites at high speed until very stiff and glossy, about 3 minutes.
Spoon the meringue into the buttered ramekins. Tap the ramekins against the work surface to settle the meringue. Run your thumb around the rim to create a small ridge between the rim and the meringue. Set the ramekins in a small roasting pan and pour in enough hot water to reach halfway up the sides of the ramekins. Cover the pan loosely with foil and bake for about 25 minutes, or until the meringues are barely set. Transfer the ramekins to a rack.
Spoon the coconut-tapioca sauce into 4 shallow bowls. Run a knife around the meringues and invert 1 into the center of each bowl. Using a tablespoon, scoop a small "cap" from the center of each meringue, fill with the mango chutney and replace the meringue "cap.". Serve the desserts immediately.
Typically, the meringues for floating island desserts are poached in milk and then set on a thin custard. In this riff, Jean-Georges cooks the meringues gently in a water bath and floats them in coconut tapioca pudding.
To keep water from spilling into the meringues, set the ramekins in a baking pan just large enough to hold them, then put the pan in the oven, and fill it with hot water. When the meringues are done, transfer them to a rack one by one using kitchen tongs.
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