Remove and discard any damaged exterior leaves from the cabbage. Cut the head in half, cut out and discard the core, and cut the cabbage into shreds. Set aside.
Trim the onion, halve it lengthwise, and peel. Place cut-side down on a cutting board and cut into thin half-moon slices.
Melt the butter in a large pot over medium-high heat. Add the onion and the salt. Cook, stirring occasionally, until the onions are wilted and softened, 3 to 5 minutes.
Add the cabbage, stir to combine, and cook, stirring occasionally, until the cabbage wilts and reduces in volume by about half, 5 to 10 minutes.
While the cabbage cooks, peel and chop the carrot; peel and chop the potatoes.
Add the carrot, potatoes, and broth. Bring to a boil before reducing the heat to maintain a steady simmer. Cook until the vegetables are completely tender, 15 to 20 minutes.
Add salt and pepper to taste and serve hot, with garnishes of your choice (or not!).