In a large bowl, combine ground beef, breadcrumbs, Parmesan cheese, egg, minced garlic, minced ginger, and chopped green onions. Season with salt and pepper. Mix until well combined.
Form the mixture into meatballs, about 1 inch in diameter.
In a large skillet over medium-high heat, add sesame oil. Once hot, add the meatballs and cook until browned on all sides, about 5-7 minutes. Remove from skillet and set aside.
In a slow cooker, combine beef broth, water, soy sauce, and brown sugar. Stir well.
Add the browned meatballs and frozen mixed vegetables to the slow cooker. Cover and cook on low for 4 hours or high for 2 hours.
About 15 minutes before serving, add the ramen noodles to the slow cooker. Stir gently to combine and allow to cook until noodles are tender.
Serve hot, garnished with additional chopped green onions.