Step 1

Position rack in upper third of oven; preheat to 450°F.

Step 2

Toss chicken, broccoli, potatoes, oil, za'atar, 1/2 teaspoon pepper and 1/4 teaspoon salt in a medium bowl. Arrange the chicken, skin-side up, and the vegetables in a single layer on a large rimmed baking sheet.

Step 3

Roast until an instant-read thermometer inserted in the thickest part of a thigh without touching bone registers 165°F and the vegetables are tender, about 25 minutes. Turn the broiler to high and broil until the chicken skin is crispy, 2 to 4 minutes.

Step 4

Meanwhile, mix cucumber, yogurt, dill, onion, lemon juice, garlic and the remaining 1/4 teaspoon each pepper and salt in a small bowl. Serve with the chicken and vegetables. Sprinkle with toasted sesame seeds, if desired.