In a large bowl, whisk together oil, lemon zest, garlic, zaatar, lemon juice, salt, and pepper. Add chicken and turn to coat. Transfer chicken and marinade to a resealable plastic bag and seal. Chill in a refrigerator or cooler.
Build a wood fire in a fire ring (see "Learn to Cook over Live Fire," below) and let it burn down to medium heat (about 350°; you can hold your hand 5 inches above cooking grate only 5 to 7 seconds).
Grill chicken, turning every 5 to 8 minutes, until well browned and cooked through, 20 to 30 minutes total; watch for flare-ups and move chicken to a cooler spot if needed. During last few minutes, grill lemon wedges and onions, turning once, just until grill marks appear. Serve with chicken.
*Find in the spice aisle of well-stocked grocery stores and at worldspice.com.
Make ahead: Through step 1, up to 1 day, chilled, or up to 2 weeks, frozen; thaw in cooler.
Wine pairing: An earthy, lighter-bodied red like La Follette Pinot Noir (North Coast), with mushroom and forest floor notes combined with violets and peppery cherry fruit that make a perfect foil for the za'atar on the chicken.