Pat chicken breasts dry with paper towels. In a bowl, combine za'atar seasoning, olive oil, and garlic cloves, salt, and pepper. Rub the mixture evenly over both sides of the chicken. Let marinate at room temperature for 20 minutes.
Place the marinated chicken on a lined baking sheet. Roast until the internal temperature reaches 165°F, about 20-25 minutes, depending on thickness. Remove from the oven, tent loosely with foil, and let rest for 5 minutes before chopping into bite-sized pieces.
While the chicken roasts, cut 2 pieces pita bread rounds into bite-sized squares. Toss with olive oil and salt. Spread in a single layer on a separate baking sheet and bake at 425°F until golden and crisp, about 8-9 minutes. Set aside to cool.
Make the sumac lemon vinaigrette. Combine lemon juice, red wine vinegar, dill, honey, sumac, and salt in a blender. Turn on the blender and slowly stream in olive oil until emulsified and well combined. Taste and adjust seasoning as needed.
Combine chopped romaine, cherry tomatoes, cucumber, feta cheese, red onion, and mint in a large bowl. Add chopped chicken and as much salad dressing as needed. Toss gently to combine.
Portion salad onto plates and top with pita chips. Serve immediately.
Notes
This recipe makes additional salad dressing. Use as much as needed.