Heat 1 tablespoon of oil in a large Dutch oven over medium-high heat. Add sausage slices and cook undisturbed for 2 minutes until deeply browned underneath. Flip and cook another 90 seconds. Remove to a plate, leaving drippings in the pot.
Add remaining oil to the pot. Add chicken pieces in a single layer and cook for 3 to 4 minutes without moving until golden. Stir and cook another 2 minutes. Remove to the same plate as the sausage.
Reduce heat to medium. Add onion, celery, and green bell pepper to the pot. Cook for 6 to 7 minutes until softened, stirring occasionally. Add garlic and stir for 60 seconds until fragrant.
Add Cajun seasoning, smoked paprika, thyme, oregano, cayenne, salt, and pepper to the vegetables. Stir for 30 seconds until the spices coat the vegetables and you can smell them blooming. Pour in the diced tomatoes and scrape up any browned bits from the bottom.
Pour in the chicken broth. Return the sausage and chicken to the pot with any collected juices. Stir in the uncooked rice to distribute evenly. Bring to a vigorous boil over medium-high heat.
Stir once from the bottom, then reduce heat to low. Cover tightly and simmer for 20 to 25 minutes without lifting the lid until the rice has absorbed the liquid and is tender.
Nestle the raw shrimp across the top of the rice. Replace the lid and cook on low for 4 to 5 minutes until shrimp are pink, curled, and opaque throughout. Scatter green onions and parsley over the top. Fold gently once and serve with hot sauce on the side.