Whisk together the egg whites, sugar, agave, lime zest, lime juice, vanilla, and salt in a double boiler or a medium heatproof bowl set over (not in) a saucepan of barely simmering water. Cook, whisking, until the mixture is hot and the sugar is dissolved, 10 to 12 minutes. Remove from the heat and stir in the flour, shredded coconut, and flaked coconut. Chill until firm, about 1 hour and up to 2 days.
Heat oven to 300° F. Line 2 baking sheets with parchment.
Roll heaping tablespoonfuls of the dough into 1-inch balls and place on the prepared baking sheets, spacing them 2 inches apart. Sprinkle with the turbinado sugar.
Bake, rotating the sheets halfway through, until golden brown, 20 to 25 minutes. Cool completely on the baking sheets.
Place the chocolate in a large microwave-safe bowl and microwave on high in 15-second intervals, stirring between each, until melted and smooth. Dip the bottoms of the cookies in the white chocolate and chill until the chocolate is firm, 15 to 20 minutes.
Storage suggestion:Keep the cookies at room temperature in an airtight container for up to 5 days.