Heat the butter in a heavy Dutch oven over medium-high heat. Add the andouille sausage and cook 4–5 minutes, stirring occasionally, until deeply browned. Remove with a slotted spoon and set aside, leaving the rendered fat in the pot.
Season the chicken with Cajun seasoning, salt, and black pepper. Brown in the same pot in batches, 3–4 minutes per side, until deep golden-brown. Remove and set aside with the sausage.
Add the onion, celery, bell pepper, and garlic to the pot. Cook 8–10 minutes, stirring constantly and scraping up browned bits, until the vegetables soften and the bottom of the pot turns deep mahogany-brown.
Pour in the warm chicken stock and scrape the bottom of the pot firmly to dissolve all the browned fond into the liquid.
Return the chicken and sausage to the pot along with the bay leaves. Bring to a gentle simmer and cook uncovered for 8–10 minutes.
Stir in the rice, making sure it's evenly distributed and mostly submerged. Bring back to a simmer, then cover tightly and reduce heat to low.
Cook covered, undisturbed, for 20–25 minutes, until the liquid is fully absorbed. Turn off the heat and let rest, covered, for 10 minutes.
Remove the lid, discard the bay leaves, and fluff gently with a fork. Taste and adjust salt, then top with sliced green onion and serve.