Remove sausage from casings and combine with breadcrumbs in a medium bowl. Form into 3/4-inch meatballs. In a large saucepot, heat oil over medium heat. Add meatballs and cook until browned all over, about 8 minutes. Drain off excess oil.
Add broth and bring to a boil over medium-high heat. Break capellini in half and stir into broth. Cook until al dente, 3 to 4 minutes. Meanwhile, in a medium bowl, whisk together vinegar, sesame oil and salt. Add cabbage, carrots and green onions and toss to combine. Ladle soup into serving bowls and top generously with cabbage mixture.