Heat a griddle pan and toast the pitta breads for 2 minutes on each side until crisp and slightly charred. Remove from the pan and rub immediately with the cut garlic. Cut the bread into squares.
Combine the pitta cubes with the green pepper, radishes, chickpeas, parsley, mint, tomatoes, onion and cucumber. Pour over the oil and lemon juice and stir in the tahini (if used). Mix until the salad is well coated. Tip into a serving dish and scatter over the sumac (if used).
Put 2 lettuce leaves on each serving plate and let people help themselves to the fatoush.