Bring a small pot of water to a boil and cook the egg for 6.5 minutes, then immediately transfer to an ice bath, peel, and slice in half.
Cook the soba noodles in a separate pot of boiling water according to package directions, drain, and set aside.
In a medium pot, heat the sesame oil over medium heat and sauté the minced garlic and grated ginger until fragrant.
Add the thinly sliced chicken breast, sea salt, and black pepper to the pot, cooking until the chicken is browned and no longer pink.
Pour in the chicken broth, tamari, and sriracha, bringing the mixture to a gentle simmer for 5 minutes.
Stir in the chopped baby bok choy and cook for 2 minutes until the leaves are wilted but the stems remain tender-crisp.
Place the cooked noodles in a deep bowl, pour the hot broth and chicken over them, and garnish with the soft-boiled egg and sliced green onions.