Make the sorbet. Scoop the melon flesh into a food processor or liquidizer and blend until smooth. Stir in the mint and add chilli to taste.
Transfer the mixture to an ice cream maker and churn until thick. Alternatively, pour the mixture into a plastic box and freeze for 4 hours, beating once or twice to break up the ice crystals.
Mix in the egg white and continue churning until the sorbet is thick enough to scoop. If you are not serving it immediately, transfer the sorbet to a plastic box and store in the freezer. Otherwise, freeze for a minimum of 2 hours until firm.
Arrange the melon quarters and ham on six serving plates. Use warm spoons to scoop out the sorbet and put two spoonfuls of sorbet on top of each melon quarter. Decorate with strips of chilli and serve immediately.