Place the chicken in a large bowl with 1 tablespoon za'atar, 1 1/2 tablespoons olive oil, and 1/2 teaspoon salt. Toss until each piece is well coated.
In a second bowl, toss the potatoes and broccoli with the remaining 1 tablespoon za'atar, 1 1/2 tablespoons olive oil, and 1/2 teaspoon salt.
Spread the chicken and vegetables in a single layer on a sheet pan, leaving space between the pieces.
Roast for 40 to 45 minutes, until the chicken reaches an internal temperature of 165°F and the vegetables are crispy at the edges.
Squeeze fresh lemon over the top and serve.
Notes
Bone-in skin-on chicken stays the juiciest, but boneless skinless breasts or thighs work too. Reduce the cook time and check the temperature early, since boneless cuts cook faster.
Za'atar is mild with no heat. For a spicier version, add a pinch of harissa or red pepper flakes to the chicken before roasting.
Swap in any sturdy roasting vegetables you like: cauliflower, carrots, sweet potatoes, or red onion all work. Keep the pieces a similar size.
To prep ahead, season the chicken and chop the vegetables up to 24 hours in advance and store covered in the fridge.