Brown the meat.
Heat olive oil in a large, heavy pot or Dutch oven over medium-high heat. Add the sliced andouille sausage and cook until lightly browned, about 3–4 minutes. Remove the sausage and set aside. Add chicken pieces and sear until golden on all sides, about 5 minutes. Don’t worry if the chicken isn’t fully cooked — it will finish in the sauce.
Sauté the vegetables.
Reduce heat to medium. In the same pot, add onion, bell peppers, and celery. Cook for 5–6 minutes, scraping up any browned bits from the bottom. Stir in garlic and cook another 30 seconds, just until fragrant.
Build the flavor base.
Add diced tomatoes (with their juices), cooked sausage, Cajun seasoning, paprika, thyme, cayenne, salt, and pepper. Stir well and let everything simmer together for 2–3 minutes so the spices bloom.
Add rice and broth.
Pour in the chicken broth and stir in the rice. Bring to a gentle boil, then reduce the heat to low. Cover and simmer for 20–25 minutes, or until the rice is tender and most of the liquid is absorbed. Stir occasionally to prevent sticking.
Finish with shrimp.
When the rice is nearly done, stir in the shrimp. Cover again and cook for 5 more minutes, or until the shrimp are pink and cooked through. Taste and adjust seasoning as needed.
Serve and garnish.
Fluff with a fork, sprinkle with green onions and parsley, and serve hot.
Tip: If the rice absorbs liquid too quickly, add a splash of warm broth or water. Jambalaya should be moist but not soupy.