How to Make It
Preheat oven to 400°F. Toss potatoes with 2 tablespoons oil, 1 teaspoon salt, and ½ teaspoon pepper on a rimmed baking sheet. Roast for 20 minutes.
Toss chicken with oregano, lemon zest, 2 tablespoons oil, ½ teaspoon pepper, and remaining 1½ teaspoons salt. Remove sheet pan from oven and push potatoes to the side. Arrange chicken on other side and return to oven. Roast until chicken is cooked through, 20 to 25 minutes. Transfer to a cutting board; let rest for 5 minutes before slicing.
Meanwhile, cook green beans in a pot of boiling salted water until bright green and crisp-tender, about 4 minutes.
Mix olives, shallot, lemon juice, and remaining 2 tablespoons oil and ½ teaspoon pepper in a small bowl. Spoon dressing over chicken. Serve with potatoes and green beans.