How to Make It
In a blender, combine the olive oil, cilantro, lemon juice, garlic, cumin, sumac, coriander and cayenne; puree until smooth. Pour the marinade into a large resealable plastic bag. Add the chicken, seal the bag and turn to coat. Refrigerate for at least 4 hours, or overnight.
Light a grill or preheat a grill pan. Remove the chicken from the marinade; reserve the marinade. Grill the thighs over a medium-hot fire, turning occasionally and brushing with the marinade until browned and cooked through, about 18 minutes. Transfer to a plate.
Preheat the oven to 300°. Bake the pitas directly on the rack for 10 minutes, or until dry and crisp but not browned. Let cool, then break into 1-inch pieces.
In a bowl, whisk the lemon juice with the vinegar and mustard. Slowly whisk in the canola and olive oils until emulsified. Season with salt and pepper.
In a large bowl, toss the lettuce, tomatoes, cucumber, parsley, purslane, mint and sumac with 1/2 cup of the dressing. Add the pita pieces and toss; add more dressing if necessary. Season with salt and pepper. Transfer the salad to a platter, top with the chicken and pass the remaining dressing separately.