Peel the mango and slice it in half lengthwise to remove the fruit from the pit. Cut each half lengthwise into thin slices and then cut the slices crosswise in half; set aside.
In a medium bowl, whisk together the flour, sugar, and baking powder. In another bowl, whisk together the coconut milk, eggs, melted butter, rum, and vanilla extract. Pour the liquid ingredients and mix with the whisk, stopping when everything is just combined. (Don't worry if the batter is a bit lumpy.) With a rubber spatula, gently but thoroughly fold in the shredded coconut.
If necessary, lightly butter, oil, or spray your griddle or skillet. Preheat over medium heat or, if using an electric griddle, set to 350 degrees F. If you want to hold the pancakes until serving time, preheat your oven to 200 degrees F.
Spoon 1/4 cup of batter onto the griddle for each pancake, allowing space for spreading. Arrange a few slices of mango on each pancake. (If you have an leftover mango slices, you can dice them and use them for garnish.) When the underside of the pancakes are golden and the tops are speckled with bubbles that pop and stay open, flip the pancakes over with a wide spatula and cook until the other sides are light brown. Serve immediately, or keep the finished pancakes in the preheated oven while you make the rest of the batch.
Place all the ingredients in a blender or a food processor fitted with the metal blade and whirl until the puree is perfectly smooth. Cover and refrigerate until ready to serve. The sauce will keep for about 4 days.