Heat vegetable oil in a large skillet over medium-high heat. Place pita pieces into the skillet without crowding. Fry in batches until golden brown and blot dry with paper towels.

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Combine cucumber, tomatoes, red onion, parsley, mint, olive oil, lemon juice, sumac, garlic, salt, and ground black pepper in a bowl. Gently toss salad with fried pita pieces. Grate feta cheese on top using a small cheese grater.