Put the sugar and cardamom pods in a heavy saucepan, pour in 600ml/1pt cold water and heat gently, stirring occasionally, until the sugar has dissolved. Increase the heat and bring to the boil, then boil for 7 minutes without stirring. Remove from the heat and leave to cool.
Meanwhile, peel the mangoes and remove as much flesh as possible from the stones. Purée the flesh in a blender or food processor, then pour into a shallow container.
Strain the cooled syrup into the mango purée, and stir well to mix. Freeze, uncovered, for 3 hours. Turn the mango mixture into a bowl, and break up the crystals with a fork.
Whisk the egg whites until stiff. Fold the egg whites into the mango mixture until evenly mixed, then return the mixture to the container. Cover and freeze overnight. Transfer to the refrigerator for 20 minutes to soften slightly before serving.