Using a serated knife cut off the tip end of each coconut. Strain 1/2 cup of the coconut liquid into a food processor bowl or blender. Using a large spoon, scoop out 2/3 cup of coconut meat; add coconut meat to food processor or blender. Cover and blend until coconut mixture is nearly smooth. Transfer to large serving bowl.
Peel, seed, and chop mangoes. Add mangoes, radishes, onions, ginger, pepper, and salt to coconut mixture and stir to combine. Refrigerate for 2 to 6 hours. Top with toasted coconut.
To toast coconut, spread coconut curls in a single layer on a baking sheet. Toast in 350 degree oven for 5 to 8 minutes, until golden brown, stirring every 2 minutes.