Prepare barley following package directions. Drain. Rinse with cold water; drain. Set aside.
Meanwhile, for tomato vinaigrette, in blender or food processor combine tomatoes, vinegar, garlic, cinnamon, 1 teaspoon black pepper, and 1/2 teaspoon salt. Blend or process until almost smooth. Blend in olive oil. Reserve 1/4 cup to brush on lamb.
Trim fat from lamb. Brush with reserved vinaigrette. Season with salt and black pepper. Place on unheated rack of broiler pan. Broil, 4 to 5 inches from heat, 10 to 15 minutes (160 degrees F); turn chops once halfway through.
In large bowl combine cooked barley, cucumber, sweet pepper, onion, and parsley. Toss with about 3/4 cup of the tomato vinaigrette. To serve, top flatbreads with greens, barley salad, and lamb chop. Pass remaining vinaigrette. Makes 4 servings.