Bring a small pot of water to a boil, add the egg, and cook for 6.5 minutes for a jammy yolk, then immediately transfer to an ice bath.
Slice the chicken breast into thin strips and season with sea salt and black pepper.
Heat sesame oil in a medium pot over medium-high heat; sear the chicken until golden brown, then remove and set aside.
In the same pot, sauté minced garlic, grated ginger, and sliced shiitake mushrooms for 2 minutes until fragrant.
Pour in the chicken broth, tamari, and sriracha, bringing the liquid to a gentle simmer.
Add the brown rice ramen noodles and chopped bok choy to the broth, cooking for 3-4 minutes until the noodles are tender.
Return the chicken to the pot to warm through.
Pour the soup into a bowl, top with the peeled and halved soft-boiled egg, and garnish with sliced green onions.