In a large Dutch oven or heavy pot, heat oil over medium-high.
Season chicken with salt and pepper and place, skin side down, in pot. Cook until golden brown on both sides, 10 to 12 minutes total. Pour off all but 1 tablespoon fat from pot; add onion and celery. Reduce heat to low, cover, and cook 20 minutes.
Add carrots, bay leaf, and 1 3/4 cups water; stir in rice and season with salt and pepper. Bring to a boil, then reduce heat to low. Cover and cook until rice absorbs almost all the liquid, 40 to 45 minutes. Let stand, covered, for 10 minutes before serving.
Serve and enjoy!