Can be made 1 day ahead (green onions) and cooled completely, uncovered, then chilled, covered. Reheat, then stir in scallion greens.
3 to 6 tablespoons oil
1 (3- to 31/2-lb) chicken, cut into serving pieces
2 1/2 teaspoons salt
1/2 cup flour
1 medium onion, chopped
1 medium green bell pepper, chopped
2 stalks celery, chopped
1 cup sliced carrots
3 cups water
1/4 teaspoon cayenne, or to taste
3/4 cup thinly sliced green onions
Cooked white rice
Heat 3 tablespoons oil in a stock pot. Pat chicken dry and sprinkle with salt. Brown chicken in 4 batches, turning, about 5 minutes per batch, transferring as browned to a large bowl. Add enough of remaining oil to pot to total 1/4 cup fat, then stir in flour with a flat metal or wooden spatula and cook over moderately low heat, scraping back and forth constantly (not stirring), until roux is the color of milk chocolate, 10 to 20 minutes. Add onion, carrots, bell pepper, and celery and cook, scraping back and forth occasionally, until onion is softened, about 8 minutes. Add water to roux mixture and bring to a boil, stirring occasionally until roux is incorporated. (Roux will appear curdled initially, but will come together as it reaches a boil.) Add chicken and any juices accumulated in bowl, then simmer, partially covered, until chicken is cooked through, 30 to 35 minutes. Stir in cayenne, green onions, and salt to taste and serve over rice.