Preheat the oven to 400°F. Set out a large oven-safe skillet and kitchen twine or toothpicks.
Brown the diced andouille over medium heat for 4–5 minutes, stirring occasionally.
Add the butter, onion, bell pepper, and celery. Cook for 5–6 minutes until softened. Stir in garlic, thyme, and ½ teaspoon Cajun seasoning; cook for 30 seconds.
Remove from heat. Fold in the cornbread or dried bread and parsley. Start with 2 tablespoons broth, adding more a tablespoon at a time until the stuffing holds together without becoming wet. Transfer to a bowl and let cool until comfortable to handle. Wipe out the skillet.
Pat chicken dry and open flat, skin-side down. Gently pound thicker areas if needed without tearing the skin. Combine the remaining 1½ teaspoons Cajun seasoning with paprika, black pepper, and garlic powder. Sprinkle one-third of the mixture inside the thighs.
Place approximately ¼ cup stuffing in each thigh. Fold the meat around it with skin outside and secure with twine or toothpicks. Avoid overfilling; bake extra stuffing separately in a greased dish. Season the outside with the remaining seasoning mixture.
Heat olive oil in the skillet over medium heat. Sear thighs skin-side down for 4–5 minutes until golden and releasing easily. Work in batches if needed. Turn skin-side up, seams underneath.
Bake uncovered for 25–35 minutes, allowing additional time for larger thighs. Confirm both the thickest part of the chicken and the center of the stuffing reach at least 165°F. Transfer to a platter and rest while making gravy.
Return the hot skillet carefully to medium heat. Leave 2 tablespoons fat in the pan, adding butter if needed, and retain the browned bits. Whisk in flour and cook 1–2 minutes. Gradually whisk in broth, scraping up browned bits. Add Worcestershire and thyme. Simmer 4–6 minutes until thickened. Taste before adding Cajun seasoning or pepper.
Remove all twine or toothpicks. Spoon gravy underneath or around the chicken to keep skin crisp. Garnish with parsley and serve with mashed potatoes or rice.