Melt 1 tablespoon unsalted butter and 2 tablespoons sugar in a large skillet and stir to melt. Add 1 ½ cups chopped mango. Cook for 2 or 3 minutes, stirring so that they caramelize. Reserve.
For the coconut cake:
Grease a 10 to 12-inch bundt cake pan (26cm) with 2 tablespoons shortening, making sure every nook, cranny and angle is covered. Dust with flour, shaking off excess. Reserve.
Preheat oven to 350ºF (180°C).
Sift 2 ½ cups all-purpose flour, 2 ½ teaspoons baking powder and ½ teaspoon salt. Set aside
In a large bowl, beat ¾ cup unsalted butter until creamy. Slowly add 1 ½ cups sugar and beat 2 minutes.
Add 3 eggs one at a time, beating well after each addition, and then beat for 2 minutes.
In a small bowl, mix 1 cup whole milk, ¼ cup coconut milk, 2 tablespoons dark rum and ½ teaspoon vanilla.
Beginning and ending with dry ingredients, add them in three parts alternating with liquids in two parts.
Add ⅔ cup shredded unsweetened coconut and stir to combine without beating.
Pour ⅔ of batter into prepared pan, spreading evenly. Scatter mango pieces on top. Cover with remaining ⅓ of batter.
Bake for 40-45 minutes, or until a toothpick or cake tester comes out clean.
Let cool on a wire rack for 15-20 minutes, then carefully invert onto the wire rack. Cool completely.
For the rum glaze:
Mix 1 cup powdered sugar with 1 tablespoon dark rum and 1 tablespoon coconut milk in a medium bowl until smooth and thick as honey. You can make it more or less thick by adjusting the amount of liquid or sugar.
Drizzle over the completely cooled cake, letting it run down the sides.
Sprinkle with 2 tablespoons shredded coconut while still wet. Let dry and serve.
Notes
Baking time: consider that all ovens and pans are different, even if they look similar. The baking time in my recipes is as accurate as it can be, but it might take you more or less time. You can use an oven thermometerto check that your oven is at the right temperature. I recommend keeping track of how your oven works and what tiny details you might need to adjust. Toasting coconut: spread it on a cookie or baking sheet and bake for about 10 minutes at 300°F (150°C). Move it around once or twice so it baked evenly. Watch it carefully so it doesn't burn.