Cook and stir ground beef in a large skillet over medium heat until crumbly, evenly browned, and no longer pink, 7 to 10 minutes. Drain excess grease, and transfer beef to a slow cooker. Place shiitake mushrooms into a food processor and pulse several times to cut the mushrooms into small pieces. Stir shiitake mushrooms, beef broth, barley, sweet onion, garlic, black pepper, salt, celery seed, and parsley with the ground beef in the slow cooker. Cook on Low for 5 hours. Stir heavy cream and green beans into the soup and cook 1 hour more.