Heat oil in a large skillet over medium-high heat. Working in batches, fry pita pieces in hot oil until golden brown on both sides. Blot dry with paper towels and set aside.
Combine cucumber, tomatoes, parsley, mint, red onion, olive oil, lemon juice, sumac, garlic, kosher salt, and ground black pepper in a large serving bowl. Gently toss salad with fried pita pieces. Grate feta cheese on top using a small cheese grater to serve.
