In a large pot, heat oil over medium-high heat. Season chicken with salt and pepper. Place chicken skin-side down in pot and brown on both sides, about 10 to 12 minutes.
Pour off all but 1 tbs fat in pot. Add onion and celery. Turn heat down to low. Cover pot and cook 20 minute.
Add carrots, bay leaf, and water. Add rice and season with salt and pepper to taste. Raise heat and bring to a boil. Reduce heat to low to bring mixture to a simmer. Cover and simmer until rice is done and all the water is absorbed, about 40 minute Let sit for 10 minutes before serving.