Heat the olive oil in a large pot over medium heat. Add the sausage and brown for about 3 minutes.
Add in the onion, celery, and green bell pepper and saute until softened, about 3 minutes. Add in the garlic and cook 1 additional minute, until fragrant.
Add in red beans, veggie broth or water, and cajun seasoning. Bring to a boil, cover, and reduce to a simmer.
Simmer on low for 3 hours*. Stirring occasionally. Checking and adjusting liquid if necessary. The beans should be soft enough to mash with a potato masher or fork.
Toward the end of cooking, mash a small section of the beans (making sure to move the sausage out of the way) with a fork or potato masher (this gives it that signature creamy consistency). Stir.
Add salt. Taste and adjust seasoning, if necessary. If your beans still aren't super soft, add 1/2-1 cup of water (if they are looking dry), cover, and continue cooking until the beans are to your liking.
Serve with white rice and garnish with green onion. Enjoy!