Step 1: Heat the vegetable oil in a large Dutch oven or heavy-bottomed pot over medium-high heat. Add the chicken and sliced andouille sausage, cooking until browned on all sides. Remove the meat with a slotted spoon and set aside.
Step 2: In the same pot, add the onion, bell pepper, and celery. Sauté for about 5-7 minutes until the vegetables are softened and the onion is translucent.
Step 3: Stir in the minced garlic, Cajun seasoning, and cayenne pepper. Cook for 1 minute until fragrant. Add the crushed tomatoes and stir to scrape up any browned bits from the bottom of the pot.
Step 4: Add the white rice and chicken broth. Return the browned chicken and sausage to the pot. Stir to combine and add the bay leaves.
Step 5: Bring the mixture to a boil, then reduce the heat to low. Cover the pot with a tight-fitting lid and simmer for 20-25 minutes without lifting the lid.
Step 6: After 20 minutes, quickly lift the lid and arrange the raw shrimp on top of the rice. Replace the lid immediately and cook for another 5 minutes, or until the shrimp are pink and the rice is tender.
Step 7: Remove from heat, discard the bay leaves, and fluff the jambalaya with a fork. Garnish with green onions and serve hot.