For the salad: Combine the chickpeas, lettuce, fennel, cabbage, bell pepper, cucumber, tomatoes, olives, onions, beans, parsley, and basil in a large bowl. Trim the ends from the lemon and discard. Cut the lemon in half lengthwise. Thinly slice the lemon and remove the seeds. Chop the lemon slices into small pieces, being careful not to lose the juice. Add the lemon pieces to the bowl. (The salad can be prepared up to 6 hours ahead. Cover and refrigerate.)
Whisk together the oil, vinegar, chiles, and garlic in a small bowl for the dressing. Add the dressing and cheeses to the salad, season generously with black pepper, and toss well. Divide the salad among dishes. Garnish with the oregano, if desired, and serve.