Trim the excess fat from the chicken pieces and sprinkle them with 1 teaspoon each of the salt and pepper.
Season 2 cups of the flour with the remaining 1 tablespoon each of salt and pepper.
Dredge the chicken pieces in the flour until coated on all sides.
Shake off any excess flour.
Heat the vegetable oil in a heavy deep skillet (cast-iron is perfect) over medium heat until the edge of a chicken piece dipped into the oil gives off a lively sizzle. Add as many chicken pieces to the skillet 4 as will fit without touching.
Fry until the pieces are browned on all sides, about 6 min utes.
Adjust the heat as necessary during frying to keep a lively sizzle without over-browning.
Remove the fried chicken to drain and repeat with the remaining pieces.
Pour off all but 4 tablespoons of drippings from the skillet. Reduce the heat to medium and add the onions, peppers, and celery to the skillet. Cook, stirring occasionally, until brown and tender, about 10 minutes.
Move the vegetables to one side of the skillet and sprinkle the 2 tablespoons of flour over the other side of the skillet. Cook the flour until golden brown, stirring constantly.
Be careful not to let the flour burn.
Slowly pour in the water and stir until the gravy is smooth.
Divide the chicken between two heavy skillets with lids or place them all in a large heavy Dutch oven.
Top with the gravy and vegetables and cover the skillets or Dutch oven tightly.
Simmer over low heat until the vegetables are tender and the chicken is cooked through, about 15 minutes.
Check the seasoning and add salt and pepper as necessary .
Serve the chicken, spooning some of the gravy and vegetables over each piece. Pass extra gravy.