Preheat the oven to 425°F with a rack in the center position.
Pat the chicken completely dry with paper towels.
In a small bowl, stir together the olive oil, 2 teaspoons of the salt, and za’atar.
On a rimmed sheet pan, toss the cauliflower and shallots with half of the za’atar mixture. Coat the chicken with the remaining mixture and arrange skin side up between the vegetables. Sprinkle the chicken all over with the remaining ½ teaspoon salt. Bake for 35 to 40 minutes, or until the internal temperature of the chicken reaches 165°F on an instant-read thermometer.
Make the za’atar oil. Meanwhile, in a small bowl, whisk together the olive oil, za’atar, salt, and garlic until combined.
Drizzle the chicken with the za’atar oil and serve with hummus, tahini sauce, tzatziki or couscous, if desired.