Combine cabbage, parsley, ¼ cup oil, and 1 teaspoon salt in bowl; set aside.
Combine yogurt, lemon juice, garlic, ¾ teaspoon salt, and ½ teaspoon pepper in second bowl; set aside.
Adjust oven rack 6 inches from broiler element and heat broiler.Line rimmed baking sheet with aluminum foil and set wire rack in sheet.
Pat chicken dry with paper towels. Combine chicken, paprika, cumin, salt, pepper, and remaining 2 tablespoons oil in large bowl.
Place chicken in single layer on prepared wire rack, smooth sides down.
Trim ends from lemon, then cut lemon in half. Place lemon halves cut side up on rack.
Broil until chicken is well browned and registers at least 175 degrees, 16 to 20 minutes, rotating sheet halfway through broiling.
Let rest for 5 minutes before slicing chicken into thin strips and transfer to platter.
Squeeze juice from 1 lemon half over chicken.
Squeeze juice from remaining lemon half into cabbage mixture and stir to combine.Transfer cabbage mixture to platter with chicken.
Arrange tomatoes and cucumber on platter. Serve with yogurt sauce and warm pita.
NOTE : Grilling is ideal!
