Preheat oven to 425 degrees.
In a small bowl, mix za'atar with 2 tablespoons oil and 2 teaspoons salt.
Rub za'atar mixture all over chicken.
Toss potatoes in silver mixing bowl with 1 tablespoon oil; season with salt and pepper. Transfer to roasting dish and roast in oven, stirring once, until they start to brown, about 20 minutes. And add chicken pieces and lemon halves. Roast until a thermometer inserted into thickest parts of chicken (not touching bones) registers about 150 degrees, 20 to 25 minutes.
Switch oven to broiler setting. Move roasting pan to upper third of oven; broil until chicken is browned, 2 to 3 minutes. In another mixing bowl, drizzle arugula lightly with oil; season with salt and pepper, and scatter over chicken and broil until just wilted and thermometer inserted into thickest parts of chicken registers 165 degrees, 1 to 2 minutes more. Let cool slightly, then squeeze lemon halves over chicken.
Toss everything together and serve. Enjoy!