Step 1: Heat 1 tbsp olive oil in a large (10-12 inch) oven-safe skillet or Dutch oven over medium-high heat. Add the sliced Andouille sausage and cook, stirring occasionally, until deeply browned and caramelized on both sides, about 5-7 minutes. Remove the sausage with a slotted spoon and set aside, leaving the rendered fat in the skillet.
Step 2: Reduce the heat to medium. Add the diced onion, green bell pepper, and celery to the skillet. Sauté for 7-10 minutes, stirring occasionally, until the vegetables are softened and translucent. Add the minced garlic and cook for another minute until fragrant.
Step 3: Stir in the Cajun seasoning, dried thyme, and smoked paprika. Cook for 1 minute, stirring constantly, to toast the spices. Stir in the tomato paste and cook for another 2 minutes, stirring to allow it to darken slightly.
Step 4: Pour in the chicken broth and diced tomatoes (undrained). Bring the mixture to a gentle simmer, scraping any browned bits from the bottom of the pan. Stir in the uncooked long-grain white rice until well combined. Return the browned Andouille sausage to the skillet, nesting it evenly. Season with salt and black pepper to taste.
Step 5: Bring the liquid back to a gentle simmer, then immediately reduce the heat to low. Cover the skillet tightly with a lid (or foil). Cook for 20-25 minutes, or until all the liquid has been absorbed and the rice is tender. Do not lift the lid during this cooking time.
Step 6: Once the rice is cooked, remove the skillet from the heat, but keep it covered. Let it rest for another 5-10 minutes. After resting, remove the lid and gently fluff the cajun sausage and rice skillet with a fork. Garnish with fresh chopped green onions and parsley before serving.