Preheat oven to 425°F / 220°C with a large sheet pan inside the oven.
Pat chicken dry and mix olive oil, yogurt, lemon juice, lemon zest, garlic, oregano, basil, salt, and pepper in a bowl. Rub over chicken and set aside.
Boil potatoes in salted water for 20 minutes until fork tender. Drain and let steam dry.
Toss potatoes with olive oil and salt. Lightly smash each potato.
Carefully place potatoes on the hot sheet pan with smashed side down. Roast for 15 minutes.
Remove tray, flip potatoes, and arrange chicken pieces between potatoes, skin side up.
Roast for 45 minutes until chicken is golden and potatoes are crispy.
Toss green beans with olive oil, garlic, and salt. Optionally boil them briefly first for a softer texture.
Add green beans to the sheet pan during the last 12–15 minutes of cooking.
Mix yogurt sauce ingredients together while the chicken finishes roasting.
Serve chicken, crispy potatoes, and green beans with yogurt cucumber sauce and fresh lemon wedges.