Combine the Chicken and Marinade ingredients in a zip-lock bag and set aside for at least 20 minutes (or preferably overnight).
Preheat oven to 350°F (180°C).
Remove chicken from marinade, but reserve the Marinade.
Heat 1/2 tablespoon olive oil in a deep, heavy skillet over medium high heat. Place the chicken in the skillet, skin side down, and cook until golden brown, then turn and cook the other side until golden brown. Remove the chicken and set aside. Pour off fat and wipe the pan with a scrunched up ball of paper towel (to remove black bits), then return to the stove.
Heat 1 tablespoon olive oil in the skillet over medium high heat. Add the onion and sauté for a few minutes until translucent. Then add rice, broth, water, oregano, salt, pepper, and reserved marinade.
Let the liquid come to a simmer for 30 seconds. Place the chicken on top, then cover skillet. Bake in the oven for 35 minutes. Remove the lid, and bake for 10 more minutes, or until all the liquid is absorbed and the rice is tender (about 45 minutes total).
(If you use chicken breasts instead of thighs, add it to the pan after 10 to 15 minutes, otherwise the breasts will overcook and become dry.)
Remove from the oven and allow to rest for 5 to 10 minutes before serving, garnished with parsley or oregano and fresh lemon zest, if desired.