Pat the chicken pieces dry with paper towels. Stir all the Cajun seasoning ingredients together in a small bowl, including the smoked paprika, dried oregano and dried thyme, then toss half of the mix with the chicken and reserve the rest.
Dice the onion, bell peppers and celery into 1 cm pieces, mince the garlic, and slice the andouille into 1 cm rounds.
Heat the olive oil in a 30 cm / 12-inch deep cast-iron skillet over medium-high heat for 2 minutes, until it shimmers. Add half the chicken in a single layer and sear for about 5 minutes, turning occasionally, until browned on both sides. Move it to a plate and repeat with the rest. The chicken will still be raw in the center.
Add the andouille to the same skillet and cook for 3 minutes, stirring now and then, until the edges darken and the fat renders. Add the onion, peppers, celery and the reserved seasoning mix with its smoked paprika, oregano and thyme. Cook for 6 minutes, scraping the bottom, until the onion turns translucent.
Stir in the garlic and tomato paste and cook for 1 minute, until the paste darkens to brick red. Pour in the tomatoes and broth, scrape up the browned bits, and bring to a simmer.
Return the chicken and any plate juices to the skillet. Simmer uncovered over medium-low heat for 12 minutes, stirring twice, until the sauce coats the back of a spoon and the thickest piece of chicken reads 165 F / 74 C.
Take the skillet off the heat and stir in the parsley and scallions. Taste before adding any salt, add hot sauce if you want more heat, and serve straight from the pan.