Make a medium-dark roux with the flour and oil over medium-high heat. After the roux has reached the right color, remove from the heat and add the onions, celery, bell pepper, and thyme. Continue to stir for a minute or so until the vegetables get soft and the roux cools.
In a large saucepan or Dutch oven, sear the pieces of chicken until browned all over. Remove the chicken. Add the wine and bring it to a boil, scraping the bottom of the pan to dissolved the browned bits in the wine.
Add three cups of water or chicken stock to the pot and bring to a simmer. Add the roux, which will be lumpy at first but can be dissolved with a wire whisk. Return the chicken to the pot.
Cover and simmer on very low heat for about a half-hour. Remove the breasts and wings, and continue to cook 20 or 30 more minutes, or until the meat on chicken legs begins to fall away from the bones. Remove the chicken from the pot and keep warm. Add the salt, pepper and Tabasco to the gravy pot and cook until the sauce thickens. You can strain it or not, according to your taste.