Rub the chicken with salt and pepper.
Heat the olive oil and butter in a pan.
Add the chicken to the hot oil, skin side down. Cook over high heat until the skins are browned. Flip and brown the opposite side. Scoop out and transfer to a plate.
To the oil in the pan (a combination of olive oil, butter and rendered chicken fat by this time), add the celery, carrot and onion. Cook over medium heat until softened.
Add the garlic, oregano, thyme and butter. Cook for another minute.
Add the rice. Stir. Pour in water (or broth). Season with salt and pepper.
Arrange the browned chicken pieces on top of the rice.
Cover the pan very tightly. Set the heat to low and cook for 30 minutes.
Sprinkle with chopped parsley before serving.