1. Preheat oven to 425 degrees, with racks in upper and lower thirds. In a small bowl, mix za'atar with 2 tablespoons oil and 2 teaspoons salt.

2. With a sharp knife, cut two 1-inch-long slashes in each piece of chicken, through skin and down to bone. Rub za'atar mixture all over chicken and into slashes.

3. On a rimmed baking sheet, toss potatoes with 1 tablespoon oil; season with salt and pepper. Roast in lower third of oven, stirring once, until starting to brown, about 20 minutes. Toss onion with remaining 1 tablespoon oil; season with salt and pepper; add to baking sheet along with chicken pieces and lemon halves. Roast until a thermometer inserted into thickest parts of chicken (not touching bones) registers about 150 degrees, 25 to 30 minutes.

4. Switch oven to broiler setting. Move baking sheet to upper third of oven; broil until chicken is browned, 2 to 3 minutes. Drizzle kale lightly with oil; season with salt and pepper. Scatter kale 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. Remove chicken to a platter; toss vegetables together and serve alongside.