Season chicken with ½ teaspoon salt and ¼ teaspoon pepper. Place flour in large bowl. Roll chicken in flour to coat, shaking off excess flour. Transfer 3 tablespoons flour to medium bowl and set aside.
In large skillet, heat oil over medium-high heat. Add chicken and cook, turning halfway during cooking, until golden brown, about 10 minutes. Transfer to plate and set aside.
Pour off all but 3 tablespoons of oil from skillet. Reduce heat to medium. Add onions, celery and garlic, and cook, stirring often, until tender, about 5 minutes. Sprinkle with reserved flour and stir well. Gradually stir in broth and bring to simmer.
Return chicken to skillet. Reduce heat to low. Cover and simmer, stirring occasionally, until chicken is cooked through and shows no signs of pink when pierced at bone, about 35 minutes. Transfer chicken to deep platter and cover with foil to keep warm. Bring sauce to boil over high heat and cook, stirring often, until thickened, about 5 minutes. Season gravy with salt and pepper and pour over chicken. Serve hot.