Preheat the oven to 425°F. Line a large sheet pan with parchment paper, if preferred.
Place the chicken thighs directly onto a sheet pan. Season each chicken thigh generously with salt and za'atar on both sides, then drizzle with oil. Toss to coat, spreading the chicken into an even layer. Let sit while prepping the other ingredients to marinate.
In a medium bowl, combine the sliced cucumber, red onion, and parsley. Drizzle with red wine vinegar and olive oil, season with salt and pepper, and toss to coat. Let this sit and marinate while the chicken cooks.
Roast the chicken for 20-25 minutes, until golden brown and cooked through (internal temperature: 165°F). In the last 5 minutes of cooking, place the naan or pita on the sheet pan to warm through.
While the chicken is cooking, mix the hummus, Greek yogurt, lemon juice and a pinch of salt in a small bowl until smooth. Taste and adjust seasoning if necessary.
Once the chicken is done, let it rest for a few minutes before cubing or rough chopping. Spread each warmed naan or pita with a dollop of hummus yogurt and spread into an even layer. Top with cucumber salad, and then add chicken on top. Finish with a squeeze of lemon. Serve immediately.