In a small bowl, mix the balsamic vinaigrette, juice from 1/2 lemon, oregano, and 1/2 teaspoon black pepper. Place chicken in a large resealable plastic bag. Pour marinade over the chicken, seal, and refrigerate for at least 1 hour
Toss the shredded cucumber with 1 teaspoon kosher salt, and allow to sit at least 5 minutes. In a medium bowl, mix the yogurt, sour cream, 1 tablespoon lemon juice, rice vinegar, and olive oil. Season with garlic, fresh dill, and Greek seasoning. Squeeze the cucumber to remove any excess water; stir into sauce. Season to taste with kosher salt and pepper. Refrigerate until ready to use.
To Freeze: Freeze in freezer safe container. Refrigerate Tzatziki sauce.
To Reheat: Remove chicken from marinade and place on prepared grill. Discard remaining marinade. Cook chicken until juices run clear, about 8 minutes on each side. Remove chicken from heat, and allow to sit about 10 minutes before slicing into thin strips. Place pita rounds on the grill, and cook for about 2 minutes, until warm, turning frequently to avoid burning. Serve with lettuce, onion, tomato, olives, feta cheese and pepperoncini if desired. Serve tzatziki sauce on side.