Scrub and slice or dice the potatoes. Shape doesn't matter since they'll be blended later. No need to peel.
Boil the potatoes in salted water until they are fork soft all the way through.
While the potatoes boil, dice the onion and finely chop the cabbage.
Drain the potatoes and leave them in the strainer.
Add the cooking oil to the pot followed by the onions, cabbage, and thyme.
Sautée the cabbage and onions until the cabbage is fully softened, stirring frequently to evenly coat with the oil and evenly sautée. About 10 minutes.
Return the potatoes to the pot with the cabbage and onions, and add the vegetable broth.
Add enough water to bring the level of liquid almost level with the vegetables in the pot.
Simmer 5-10 minutes.
While the soup is simmering, blend together the cannellini beans (with the aquafaba - liquid from the can), nutritional yeast, and butter until smooth.
OPTIONAL: For a chunkier soup, remove some of the cabbage and potatoes from the pot with a slotted spoon and set aside. Stir back into the soup after blending.
Pour the blended bean cream into the simmering soup and use an immersion blender to blend it all together until reaching the desired consistency. I like to leave mine just a little bit chunky, but others prefer it completely smooth - especially if you have picky eaters in the house! If necessary, add more water during blending to thin out the soup.
Season to taste with black pepper and salt.