Bring a small pot of water to a boil. Carefully lower the egg into the water and boil for 6.5 minutes. Immediately transfer to an ice bath, then peel and set aside.
Cook the soba noodles in the same boiling water according to package directions. Drain and rinse with cold water.
In a medium pot, heat the toasted sesame oil over medium heat. Add the sliced chicken breast, sea salt, and black pepper, sautéing until browned.
Stir in the minced garlic, grated ginger, and sliced shiitake mushrooms. Cook for 2 minutes until fragrant.
Pour in the chicken broth, tamari, and sriracha. Bring to a gentle simmer.
Add the chopped baby bok choy to the broth and cook for 2 minutes until the greens are wilted but the stems remain crisp.
Place the cooked soba noodles in a deep bowl and pour the hot broth, chicken, and vegetables over them.
Slice the soft-boiled egg in half and place on top. Garnish with thinly sliced green onions before serving.