In a small saucepan, combine the coconut water and sugar and bring to a boil. Cook until reduced to 3/4 cup, about 6 minutes. Let the sugar syrup cool.
In a blender or food processor, puree the diced mango with 2 tablespoons of the sugar syrup. Transfer 1/4 cup of the puree to a bowl and whisk in the Greek yogurt, coconut milk, lemon juice and the remaining sugar syrup. Pour the mixture into a glass baking dish and freeze for 1 hour, until frozen around the edges. Whisk the mixture to break up the clumps and return to the freezer. Freeze for about 2 hours longer, whisking frequently, until the mixture is nearly frozen. Spread the remaining mango puree on top and, using a butter knife, swirl it into the yogurt. Freeze until nearly solid.
Meanwhile, in a small skillet, toast the shredded coconut over moderate heat, stirring constantly, until it’s golden, about 3 minutes. Transfer the coconut to a plate and let it cool. To serve, scoop the frozen yogurt into bowls and top with the toasted coconut. Alternatively, in a tall glass, layer the coconut between small scoops of the frozen yogurt, like a parfait.