Bring a small pot of water to a boil, carefully lower the egg into the water, and simmer for 6 minutes and 30 seconds.
Immediately transfer the egg to an ice bath to stop the cooking process; once cool, peel and set aside.
Cook the soba noodles in a separate pot of boiling water according to package directions, then drain and rinse with cold water.
Heat sesame oil in a medium saucepan over medium heat and add the chicken, garlic, and ginger, sautéing until the chicken is golden and cooked through.
Pour in the chicken broth, tamari, and sriracha, then bring the mixture to a light simmer.
Add the shiitake mushrooms and baby bok choy to the broth, cooking for 2-3 minutes until the vegetables are tender yet vibrant.
Place the noodles in a bowl and pour the hot broth, chicken, and vegetables over them.
Slice the soft-boiled egg in half and place it on top, finishing with a garnish of sliced green onions.