Combine the chicken, quartered onion, quartered celery stalk, bay leaf, fresh thyme and peppercorns in a large pot or Dutch oven.
Fill the pot with water, covering the chicken, or as close as you can get to covering it. (You will want about 7-8 cups of stock for the gumbo).
Heat the pot to boiling, then reduce to a simmer.
Cover and simmer for 2 hours.
Remove the chicken and set aside to cool. Shred it once cool to the touch.
Strain the stock and set aside.
Heat the flour and oil over medium heat, whisking constantly.
Continue whisking until it becomes a deep medium brown color (like milk chocolate). It will take 10-15 minutes.
Add the onion, green pepper, celery and andouille sausage.
Cook, stirring regularly, for about 3-5 minutes.
Add the seasoning (Cajun seasoning, salt, chili powder and thyme) and cook another 3 minutes.
SLOWLY stir in the 7-8 cups of stock, whisking constantly to avoid clumping. ** I recommend stirring in 1 cup at a time.
Simmer for 10 minutes, then stir in about 2 cups of the shredded chicken (add more or less as desired).
Add additional salt, if needed to taste.
Serve with steamed white rice and green onions.