1½ cups unsweetened large flaked coconut
2½ cups whole milk
½ cup large-pearl tapioca
1 ripe mango
2½ tablespoons granulated sugar, plus more to taste
¼ teaspoon vanilla extract
1/8 teaspoon coconut extract (optional)
1/8 teaspoon kosher salt
1. Preheat the oven to 350°. Add the coconut flakes to a rimmed baking sheet and toast in the oven until the flakes are golden-brown and fragrant, about 5 minutes. Remove the pan from the oven and transfer the coconut flakes to a large plate to cool completely.
2. In a medium saucepan set over medium heat, add the whole milk. Once the milk comes to a soft boil, turn off the heat and add 1 cup of the toasted coconut flakes. Set aside to steep for 10 minutes, then pour the coconut-milk mixture into a fine-mesh sieve set over an medium bowl. Use a rubber spatula to press on the coconut flakes to extract as much liquid as possible. Discard the coconut and pour half of the coconut-infused milk into an airtight container. Add the tapioca, cover and refrigerate overnight. Refrigerate the remaining liquid in a separate airtight container.
3. Slice the bottom off of the mango, then peel it using a vegetable peeler. Stand the mango upright and slice off the two fleshy, wide lobes on either side of the pit. Carve away the easy-to-remove fruit from around the pit and chop the fruit into ¼-inch pieces. Set one-quarter of the diced mango aside and add the remaining mango to a food processor. Process the mango until totally smooth (if the mango proves difficult to purée, add up to 2 tablespoons of water to help it along).
4. Transfer the tapioca mixture to a medium saucepan and whisk in the remaining coconut-infused milk. Bring to a boil over medium heat, then add the sugar, vanilla, coconut extract (if using) and salt. Reduce the heat to low and simmer, stirring constantly, until the pudding is thick, about 10 minutes. Gently swirl in the puréed mango until the mixture looks marbled.
5. Spoon the tapioca into 4 small glasses, bowls or ramekins. Serve warm or refrigerate until the tapioca is chilled and set, at least 4 hours. Serve topped with the reserved chopped mango and the remaining ½ cup toasted coconut flakes.