Place the beef chuck roast pieces 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 the miso paste is dissolved.
Pour the broth mixture over the beef in the slow cooker.
Cover and cook on low for 6 hours, or until the beef is very tender.
About 30 minutes before serving, remove the beef from the slow cooker and shred it using two forks.
Return the shredded beef to the slow cooker and add the sliced shiitake mushrooms, julienned carrots, and ramen noodles.
Cover and cook on high for 20 minutes, or until the noodles are tender and the vegetables are cooked.
While the noodles cook, bring a small pot of water to a boil. Carefully add the eggs and boil for 7 minutes for soft-boiled eggs.
Remove the eggs and place them in a bowl of ice water for 5 minutes. Peel and cut each egg in half.
Stir the baby spinach leaves into the slow cooker and cook for 2 minutes until wilted.
Add rice vinegar to the soup and stir to combine.
Ladle the ramen noodle soup into bowls. Top each bowl with two halves of soft-boiled egg, sliced green onions, and a sprinkle of toasted sesame seeds.
Serve immediately.