Cook ground beef in a 6-quart Dutch oven over medium heat, breaking up chunks with a wooden spoon, until browned, 6 to 8 minutes. Remove from Dutch oven with a slotted spoon.
Cook andouille sausage in the Dutch oven over medium-high heat, stirring occasionally, until browned, 8 to 10 minutes. Remove sausage with a slotted spoon. Drain all but 1 tablespoon drippings from the Dutch oven.
Add pepper, celery, and green onions to the Dutch oven. Cook, stirring, over medium heat until tender, 5 to 6 minutes. Stir in garlic, tomato paste, Cajun seasoning, and salt. Cook and stir until fragrant, about 1 minute.
Add cooked ground beef, sausage, cabbage, and broth to the Dutch oven. Bring to a boil, then reduce heat to medium-low. Cover and simmer until flavors combine and cabbage is very tender, about 25 minutes.
Stir in vinegar. Garnish with additional green onion.
