Heat a tablespoon of oil in a large Dutch oven over medium heat. Add the andouille and brown on all sides. Remove to a paper towel-lined plate.
Season the chicken thighs with salt and pepper and add to the pot. Brown on both sides, about 15 minutes. Remove to a paper towel-lined plate.
Add the butter and melt. Whisk in the flour and cook 10-12 minutes or until brown.
Add onion, celery, bell pepper and garlic to a food processor. Pulse a few times to roughly chop then add to the Dutch oven with the roux. Stir and cook for about 5 minutes to cook out some of the liquid.
Add the beer and deglaze the pot. Make sure to scrape up the yummy bits that may have stuck to the bottom.
Add the chicken stock, bay leaves, paprika and cayenne. Stir.
Add in the chicken pieces and simmer for 1 hour or until the chicken has started to fall off the bones.
Carefully remove the chicken to a cutting board and pull the meat from the bones. Shred and return to the pot.
Slice the andouille and add back to the pot. Warm thoroughly.
Serve over warm cooked rice garnished with chopped parsley and green onions.