Combine the olive oil, lemon juice, garlic, za’atar, and salt in a large bowl. Add the chicken, turn to coat and let sit at room temperature for 10 minutes, while you preheat the grill.
Place chicken and onions on the preheated grill and cook with lid open for 5-6 minutes per side, or until chicken is cooked through and nice grill marks appear. About 2 minutes before the chicken is done, grill the halved lemons, cut side down.
Place arugula into a large mixing bowl. Next, roughly chop the olives and add to the bowl as well. Halve the cherry tomatoes and add them to the bowl. Tear the basil leaves and oregano and add to the mixture. Crumble in the feta cheese and sprinkle with salt and pepper.
Remove chicken, onions and lemons from the grill. Roughly chop the onions and add to the salad. Juice one lemon half over the salad. Add a drizzle of olive oil and toss everything together, then transfer to a plate.
Thinly slice the chicken and add to the salad. Squeeze remaining lemon half over the chicken. Garnish with grilled lemon halves.