Cut chicken into 1-inch cubes, slice sausage, dice vegetables, mince garlic, and measure all spices. Refrigerate shrimp until needed.
Heat oil in Dutch oven over medium-high heat. Season chicken with salt and pepper, brown for 3-4 minutes per side, then remove and set aside.
Cook andouille sausage in the same pot for 4-5 minutes until browned and caramelized. Remove and set aside with chicken.
Reduce heat to medium. Add onion, bell pepper, and celery to pot. Cook for 8-10 minutes, stirring occasionally, until vegetables are soft and onions are translucent.
Add garlic and cook for 1-2 minutes until fragrant. Pour in diced tomatoes with juices, scraping up browned bits from bottom of pot. Cook for 2-3 minutes.
Stir in all seasonings and cook for 1 minute. Add chicken broth and bay leaves. Return chicken and sausage to pot with any accumulated juices. Bring to a boil.
Once boiling, stir in rice, ensuring it's evenly distributed and submerged. Reduce heat to low, cover tightly, and simmer undisturbed for 20 minutes.
Remove lid and nestle raw shrimp into rice mixture. Cover and cook for 5-7 minutes until shrimp are pink and opaque.
Turn off heat and let rest covered for 5 minutes. Remove bay leaves, fluff with fork, adjust seasoning, and garnish with green onions and parsley. Serve with hot sauce.