Fill a medium pot with 5 cups of water and bring to a boil. Reduce to medium heat, add the tapioca pearls, and cook for 10 minutes, stirring occasionally and keeping the water bubbling throughout. Turn off the heat, cover with a lid, and let rest on the warm burner for 10 to 15 minutes. If white dots remain in the centers, keep covered and let rest for another 2 to 3 minutes at a time until the pearls are completely translucent.

Pour the pearls into a fine-mesh sieve to drain. Rinse thoroughly under cold running water until completely cool to stop the cooking and wash away excess surface starch. Drain well, then transfer the tapioca pearls to a large mixing bowl.

Peel, pit, and slice two of the mangos, then transfer them to a blender with the condensed milk. Blend on high until completely smooth. Set aside.

Slice the cheeks from the remaining mango as close to the pit as possible. Score the flesh into 1-cm cubes, then use a large spoon to scoop the mango away from the skin. Set aside.