Preheat oven to 375°F. Grease a 9x13-inch baking dish.
Heat olive oil in a large skillet over medium. Brown sausage until lightly crisp, 5–6 minutes.
Add onion, bell pepper, and celery. Cook until softened, 4–5 minutes. Stir in garlic for 30 seconds.
Season with paprika, thyme, salt, and pepper. Remove from heat.
Spread uncooked rice in the baking dish. Spoon sausage mixture evenly over the rice.
Pour in broth and tomatoes with juices. Dot with butter, then stir to combine evenly.
Cover tightly with foil. Bake 40 minutes.
Uncover, stir well, and check rice. Bake 10–15 minutes more until tender and mostly absorbed.
Scatter cheddar on top. Bake 5 minutes to melt, then rest 5 minutes.
Garnish with parsley and green onion. Fluff gently and serve hot.