Peel onions, leaving roots intact; cut each onion into 4 wedges. Trim fat from brisket, and thinly slice. Combine brisket, onions, water, and next 9 ingredients (water through turnips) in a large Dutch oven or stockpot; bring to a boil. Reduce heat, and simmer, uncovered, 2 hours or until brisket is tender. Cover and simmer 1 hour. Strain mixture through a cheesecloth-lined colander into a large bowl; discard garlic, bay leaves, parsley, and leeks. Cover broth, and chill at least 24 hours. Place brisket, onions, carrots, celery, and turnips in a bowl; cover and chill.
Skim solidified fat from surface of broth, and discard. Combine broth and brisket mixture in Dutch oven, and cook over medium heat until thoroughly heated.
Ladle soup into large shallow bowls, and top with Green Sauce. Serve with parslied potatoes.