Using scissors, cut the cartons from the mango sorbet. Lay the sorbet on its side and cut each pint into 4 rounds. Arrange the sorbet slices on a plate lined with plastic wrap and freeze until firm.
In a small saucepan, combine the sugar and water and bring the mixture to a boil. Add the basil leaves and blanch just until wilted, about 30 seconds. Let cool slightly, then transfer the mixture to a blender and puree until smooth. Let the puree cool. In a medium bowl, stir the basil puree into the vanilla ice cream; leave visible streaks in the ice cream. Freeze the basil ice cream until firm.
Preheat the oven to 225°. Arrange racks in the top, middle and bottom of the oven and line 3 large baking sheets with parchment paper. In a medium bowl, whisk the egg whites and confectioners' sugar until smooth. Set the bowl over a pot of simmering water and whisk until warm, about 5 minutes. Remove from the heat and beat at high speed until soft peaks form. Beat in the lime zest.
Spoon the lime meringue into a pastry bag fitted with a 1/2-inch round tip and pipe 1/2-inch kisses onto the prepared baking sheets, about 1/2 inch apart. Bake for about 1 1/4 hours, until the lime meringues are firm and dry; shift the pans among the racks 3 times for even baking. Turn off the oven, prop the door open and let the meringues cool completely, about 1 hour.
In a medium bowl, beat the heavy cream with the sugar until soft peaks form. Beat the grated lime zest into the whipped cream.
Arrange the frozen mango sorbet slices in the bottoms of shallow bowls or on dessert plates. Top with scoops of the basil ice cream. Top with some of the lime meringue kisses and dollops of the lime whipped cream. Garnish with diced mango, basil leaves and more lime meringue kisses. Serve the vacherin right away.