Bring a small pot of water to a boil, carefully lower the egg into the water, and simmer for 6.5 minutes for a jammy yolk; immediately transfer to an ice bath.
In a medium pot, heat the toasted sesame oil over medium heat and sauté the minced garlic, grated ginger, and sliced shiitake mushrooms until fragrant and softened.
Pour in the chicken broth, tamari, and sriracha, bringing the liquid to a gentle simmer.
Add the thinly sliced chicken breast to the simmering broth and cook for 4-5 minutes until fully opaque and tender.
Stir in the soba noodles and chopped bok choy, cooking for an additional 3-4 minutes until the noodles are al dente and the greens are vibrant.
Season the soup with sea salt and black pepper to taste.
Peel the cooled soft-boiled egg and slice it in half lengthwise.
Ladle the hot soup into a large bowl, top with the egg halves and sliced green onions, and serve immediately.