Set a Dutch oven over medium heat and add a 2-count of olive oil. Add the sausage links and brown slightly as you render some of the fat. Remove and set aside on a paper towel. Season chicken with plenty of salt and pepper and add skin side down to the pan. Cook over medium heat to render fat for 7 to 10 minutes, then turn to brown both sides of the chicken. Remove and set aside on a paper towels.
With the pot still over medium heat add 2 tablespoons butter to melt with the fat then add flour and whisk to incorporate, and then swap to a wooden spoon. Cook until it is nice and brown (you want a nice deep rich colored roux), about 10 to 12 minutes.
In a food processor add the onion, celery, bell pepper, and garlic, and pulse to roughly chop. Add the vegetable pulp to the pot with the roux and stir with a wooden spoon. Cook over medium heat for 5 to 7 minutes to sweat out some of the moisture. Add the stock, paprika and cayenne. Return the chicken pieces to the pot and simmer until the chicken is tender, about 1 hour. Remove the chicken pieces from pot to a cutting board and shred the meat. Discard the bones and return the meat to the pot. Cut the andouille sausage into chunks and nestle back into the pot amongst chicken to heat through.
Give it all a final taste and season with salt and pepper, to taste. Transfer to a serving platter and garnish with plenty of parsley and scallions.