Warm the olive oil in a large pot over medium heat. Add the onion, carrots, celery and garlic and sauté until the onions are translucent, about 8 minutes.Add the herbs de provence and stir.Add the stock, barley, salt and pepper and bring to a boil. Lower to a simmer and cook partially covered until the barley is tender, at least 30 minutes.Stir in the cooked turkey and simmer until turkey is heated through. If you want the barley to become larger and thicken the soup a little, just simmer another 10-20 minutes.Stir in the lemon juice and spinach and cook until spinach is just wilted. Taste for seasoning and add additional salt, pepper or lemon juice, if needed. Serve with grated Pecorino or Parmesan cheese, if desired.