Add the chicken, onion, garlic, broth, soy sauce, mirin, coconut milk, mushrooms, ginger, and black pepper to a 6-quart (or larger) slow cooker.
Cover and cook on LOW for 4–6 hours, until the chicken is tender and the broth is flavorful.
Remove the chicken and shred or slice it, then set aside.
Cook the ramen noodles separately in boiling water according to the package directions. Drain and set aside.
For the soft-boiled eggs, bring about 2 inches of water to a boil in a covered pot. Once boiling, gently lower the eggs in using a slotted spoon and boil for 6 minutes.
Transfer the eggs to an ice bath for a minute or two, just until cool enough to handle, then peel.
To serve, divide the noodles between bowls and ladle the hot broth over the top. Add the shredded chicken, halved soft-boiled eggs, and any toppings you like.