Place the beef chuck roast chunks into the slow cooker.
In a medium bowl, whisk together beef broth, water, soy sauce, miso paste, sesame oil, grated ginger, minced garlic, brown sugar, and crushed red pepper flakes until well combined.
Pour the broth mixture over the beef in the slow cooker.
Add the sliced carrots and shiitake mushrooms on top.
Cover and cook on low for 6 hours until the beef is tender.
About 20 minutes before serving, bring a small pot of water to a boil. Gently add the eggs and boil for 7 minutes for soft-boiled eggs.
Remove eggs from boiling water and place in an ice bath to cool. Peel and set aside.
Add the dried ramen noodles to the slow cooker and stir gently. Cover and cook on high for 5-7 minutes until noodles are tender.
Stir in the baby spinach leaves and sliced green onions. Let cook for 2 more minutes until spinach is wilted.
Divide the soup into bowls. Slice the soft-boiled eggs in half and place on top of each bowl.
Serve immediately.