Put the onion in a bowl, pour over enough boiled water from a kettle to cover, then set aside for 5 mins. Drain and rinse under cold water. Toss with the olives, cucumber, tomatoes, parsley and/or mint leaves, beetroot and chicken.
Season the yogurt and toast the flatbreads. To serve, top each flatbread with generous spoonfuls of the souvlaki, yogurt and a drizzle of oil.