Heat the olive oil in a large skillet or Dutch oven over medium-high heat. Add the sausage slices in a single layer and cook, stirring occasionally, until they are browned on both sides, about 5 minutes. Use a slotted spoon to transfer the sausage to a plate.

Reduce the heat to medium. Add the onion, bell pepper, and celery to the same skillet. Cook, scraping up any browned bits, until the vegetables are softened, about 5 minutes. Add the garlic and cook for another minute, until fragrant.

Stir in the rice and cook for about 1 minute, stirring constantly. This toasting step adds a nutty depth to the dish.

Pour in the chicken broth, diced tomatoes, Cajun seasoning, thyme, paprika, oregano, and a pinch of black pepper. Stir everything together. Bring the mixture to a boil, then reduce the heat to low. Cover the skillet and let it simmer for 18 to 20 minutes, or until the rice is tender and most of the liquid is absorbed.

Once the rice is cooked, gently stir the browned sausage back into the skillet. Add the butter if using, and stir until it melts. Taste and adjust salt and pepper. The rice should be fluffy and the sausage should be heated through.

Remove the skillet from the heat and let it sit, covered, for 5 minutes. This allows the flavors to meld. Then fluff the rice with a fork, sprinkle with fresh parsley or green onions, and serve with hot sauce on the side.