Heat oil in a large pot over medium heat. Add leeks, carrot, celery and thyme.
Cover and cook, occasionally stirring, for 10 minutes.
Add broth (or stock), lager, cabbage, potatoes and bay leaves; cover and bring to a boil.
Uncover, reduce heat and simmer for 10 minutes.
Stir in greens and sausage and cook, occasionally stirring, until the potatoes are tender, 10 to 15 minutes more.
Season with salt and pepper. Discard bay leaves.
Ladle into bowls and top each with parsley, if desired.