In a large slow cooker, combine beef chuck roast pieces, beef broth, water, soy sauce, miso paste, grated ginger, minced garlic, sesame oil, brown sugar, rice vinegar, and crushed red pepper flakes. Stir gently to combine.
Cover and cook on low for 6 hours, or until the beef is tender and easily shredded with a fork.
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.
Add the sliced shiitake mushrooms, julienned carrots, and half of the chopped green onions to the slow cooker. Stir well and cover.
In a separate pot, bring water to a boil and cook ramen noodles according to package instructions, about 3 minutes. Drain and set aside.
Meanwhile, bring a small pot of water to a boil. Carefully add the eggs and boil for 7 minutes for soft-boiled eggs. Remove eggs and place in ice water to cool. Once cooled, peel and slice eggs in half.
Add the baby spinach to the slow cooker and stir until wilted, about 2 minutes.
To serve, divide cooked ramen noodles into four bowls. Ladle the beef and broth mixture over the noodles.
Top each bowl with sliced soft-boiled eggs, remaining chopped green onions, and toasted sesame seeds.