Heat oil over medium high heat in a cast iron or heavy bottomed Dutch oven. Season chicken on both sides with the salt, pepper and Cajun seasoning. Cooking in batches, brown the chicken on all sides; remove and set aside.
Add oil to the pan drippings, if needed, to equal 1/2 cup total and heat over medium high heat, slowly whisking in the flour a little at a time. Cook until flour is incorporated and no lumps remain. Cook, stirring regularly, for 4 minutes.
Add the onion, bell pepper and celery and cook for 3 minutes, then begin adding in warmed water or broth, a cup at a time, constantly whisking it in until well incorporated and blended in.
Bring to a boil, reduce heat to low simmer, return the chicken to the pot and simmer uncovered for 1 hour. Add the chopped garlic and hot sauce and let cook another 30 minutes. Taste for seasonings and add additional salt, pepper and Cajun seasoning as needed.
To thicken the drippings for a gravy, skim excess fat from the top of the drippings and discard. Prepare a slurry of 2 tablespoons of cornstarch and just enough water to dissolve.
Remove the chicken from the pot and set aside. Bring drippings up to a boil, slowly stirring in the slurry. Boil, stirring constantly until mixture is thickened to desired consistency.