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.
Preheat a barbecue for high heat.
Toss the grated cucumber with 1 teaspoon sea salt and allow to sit at least 5 minutes. In a medium bowl, mix the yoghurt, 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 sea salt and pepper. Refrigerate until ready to use.
Remove chicken from marinade and place on prepared barbecue. 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 bread on the barbecue and cook for about 2 minutes until warm, turning frequently to avoid burning. Arrange warmed pita, sliced chicken, lettuce, onion, tomato, olives and pepperoncini on a serving plate. Serve tzatziki sauce and feta cheese in separate bowls on the side. Stuff pita pockets with chicken and toppings to serve.