Heat a Dutch oven over medium-high heat. Add half the andouille sausage pieces. Brown sausage, stirring occasionally until seared on all sides (no need to cook through), about 8 minutes. Remove from the pot, and then repeat with the remaining sausage. Set sausage aside.
Add the avocado oil (if your sausage released a lot of fat, you might not need to add the oil). Then add the celery, bell pepper, onion, and garlic. Cook until just tender and flavorful, about 6 minutes.
Add the sausage pieces, chicken stock, diced tomatoes, rice, Cajun seasoning, bay leaves, smoked paprika, and thyme. Bring to a boil, reduce heat to a simmer, and simmer with lid ajar, stirring occasionally for 20-25 minutes, or until the veggies are tender and the rice is soft.
Add in the cooked shrimp, then serve the jambalaya topped with the green onions.