Heat the oil in a Dutch oven or large pot over medium-high heat until shimmering. Add the sausage and cook until browned, about 6 minutes. Using a slotted spoon, remove the sausage to a plate and set aside.
Reduce the heat to medium. Add the onion, celery, and bell pepper to the pot and cook, stirring occasionally, until softened, 8 to 10 minutes. Add the rice, garlic, bay leaf, salt, oregano, thyme, black pepper, and cayenne. Cook, stirring occasionally, until the rice makes a crackling/sizzling sound, about 3 minutes.
Add the broth, water, and reserved sausage. Scrape up any browned bits from the bottom of the pot. Bring to a boil. Stir the rice and spread into an even layer. Reduce the heat to low, cover, and simmer for 10 minutes. Stir the rice, cover again, and cook undisturbed until the rice is tender, 5 to 10 minutes more. All the liquid should be absorbed.
Remove from the heat and let it sit covered for 5 minutes for the rice to steam. Fluff the rice with a fork and season with more salt and pepper as needed. Sprinkle with the green onions and serve.