Peel and seed 3 mangoes. Chop the mango and measure out about 2 cups of mango. Transfer the mango to a heavy duty blender or food processor and add the lime juice, corn syrup and salt. Blend or process until smooth. If the mango is particularly pulpy, pour the puree through a strainer to remove any fibrous pieces. You should have about 1 3/4 -2 cups of mango puree.
Add egg yolks to a medium mixing bowl. Beat sugar into the egg yolks until thick and pale yellow. Beat in the cornstarch and set aside.
Bring the milk, coconut milk and heavy cream to a simmer in a medium saucepan. Remove from heat and whisk the hot milk into the egg mixture in a slow, steady stream until well combined. Pour the milk mixture back into the saucepan. Heat over low heat stirring constantly with a wooden spoon or whisk until mixture thickens. Don't let it boil or the eggs will scramble. Remove the custard from the heat. Set a strainer over a large bowl and pour the custard through the strainer to remove any egg solids. Let the custard cool slightly. Stir in the coconut extract. Add the mango puree and stir to combine. Cover the custard with plastic wrap and refrigerate until very cold or overnight.
Stir the chilled custard before proceeding.
Preheat the oven to 300 degrees. Place the flaked coconut onto a baking sheet in a single layer. Bake for 6-10 minutes until coconut is lightly browned. Remove from oven and set aside.
Peel and seed the remaining mango and dice into 1/4" cubes - you should have about 1 cup of fruit.
Freeze the ice cream according to your ice cream maker's manufacturer's instructions. When the ice cream is semi frozen, add the toasted coconut and mango chunks to the ice cream and allow the machine to mix in the ingredients.
When finished the ice cream will be soft and ready to eat. For firmer ice cream, transfer to a freezer safe container and freeze at least two hours or overnight.