Season the chicken with 1/2 teaspoon salt and 1/4 teaspoon pepper. Place the flour in a large bowl. Roll the chicken in the flour to coat, shaking off excess flour. Transfer 3 tablespoons of the flour to a medium bowl and set aside.

In a large skillet, heat the oil over medium-high heat. Add the chicken and cook, turning halfway during cooking, until golden brown, about 10 minutes. Transfer to a plate and set aside.

Pour off all but 3 tablespoons of the oil from the skillet. Reduce the heat to medium. Add the onions, celery, and garlic and cook, stirring often, until tender, about 5 minutes. Sprinkle with the reserved flour and stir well. Gradually stir in the broth and bring to a simmer.

Return the chicken to the skillet. Reduce the heat to low. Cover and simmer, stirring occasionally, until the chicken is cooked through and shows no sign of pink when pierced at the bone, about 35 minutes. Transfer the chicken to a deep platter and cover with foil to keep warm. Bring the sauce to a boil over high heat and cook, stirring often, until thickened, about 5 minutes. Season the gravy with salt and pepper and pour over the chicken. Serve hot.