Prepare the shortcakes: Line a 9-by-13-inch baking sheet with parchment paper. In a medium bowl, combine the flour, ½ cup sugar, ¼ shredded coconut, baking powder, baking soda and salt, and whisk to incorporate. Working quickly and using your fingers or a pastry cutter, rub the cold butter into the dry mixture. Cut the butter into the flour until the pieces are the size of small pebbles. (Alternatively, use a food processor to pulse the dry ingredients with the butter.)

Make a well in the center of the bowl. Add the egg yolk, stirring with a wooden spoon to incorporate. Add the buttermilk a tablespoon at a time until a shaggy dough forms.

Transfer the dough to a lightly floured work surface and press the clumps together until they form a slightly uniform piece. Lightly dust the top of the dough with flour. Pat or roll the dough into an 8-inch square and, using a sharp knife or bench scraper, cut into 12 squares. Transfer the shortcakes to the prepared baking sheet, cover and chill the squares in the refrigerator for at least 30 minutes and up to 12 hours.

Heat oven to 350 degrees. In a small bowl, whisk the reserved egg white with 2 tablespoons room temperature water. Brush the dough with the egg white-water mixture and sprinkle with sugar. Bake until cooked through, rotating the baking sheet once, about 20 to 25 minutes. Remove the shortcakes from the oven and allow to cool completely on a rack set inside a baking sheet.

Prepare the filling: In a medium bowl, combine the mango with ¼ cup sugar. Using a grater or Microplane, zest the lime right into the bowl and add the juice from the lime. Add a pinch of salt and the black pepper. Combine with a spoon or spatula, and set aside or refrigerate.

Make the whipped cream: On a parchment-lined sheet pan, scatter ¼ cup coconut and toast in the 325-degree oven for 3 minutes. Cool completely.

To the bowl of a standing mixer fitted with a whisk attachment, add the heavy whipping cream and ¼ cup sugar. Mix on medium-high speed until smooth and fluffy, about 1 to 2 minutes. Add the toasted coconut to the whipped cream, and mix it in gently.

Serve: Once completely cooled, cut each shortcake in half crosswise. Top the bottom halves with 2 spoonfuls of mango. Drizzle over some syrup from the bowl, and top with coconut whipped cream and the other half of the shortcakes.