Bring a small pot of water to a boil, carefully lower the egg into the water, and cook for 6.5 minutes.
Immediately transfer the egg to an ice bath for 5 minutes, then peel and slice it in half.
Season the chicken breast with sea salt and black pepper on both sides.
In a large pot over medium heat, sear the chicken breast until golden and cooked through, then remove from the pot and slice into thin strips.
In the same pot, add the toasted sesame oil, minced garlic, and grated ginger, sautéing for 1 minute until fragrant.
Pour in the chicken bone broth and tamari, scraping the bottom of the pot to release any browned bits, and bring to a gentle simmer.
Add the brown rice ramen noodles and sliced shiitake mushrooms to the broth and cook for 3 minutes.
Stir in the chopped baby bok choy and cook for an additional 1 minute until the leaves are wilted but the stems remain crisp.
Ladle the broth, noodles, and vegetables into a deep bowl.
Top the soup with the sliced chicken, the soft-boiled egg halves, and thinly sliced green onions before serving.