Whisk olive brine, 1 1/2 tsp oregano, and mustard in medium bowl. Slowly whisk in 4 tbsp oil until emulsified. Add cucumbers, olives, and half of onion and toss to combine. Season with salt & pepper; set aside.
Combine chicken, feta, remaining onion, 1 tbsp oil, and remaining oregano in large bowl. Cover half of each pita round with one-quarter of chicken mixture and fold over. Brush both sides of pita with remaining oil.
Heat grill pan or large nonstick skillet over medium heat for 1 min. Place 2 sandwiches in pan and weight with Dutch oven. Cook sandwiches until golden brown and cheese is melted, about 5 mins per side. Repeat with remaining sandwiches. Serve with cucumber-olive salad.
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