Bring a small pot of water to a boil, carefully lower the egg in, and simmer for 6.5 minutes for a jammy yolk, then transfer immediately to an ice bath.
In a separate medium pot, cook the soba noodles according to package directions, drain, and set aside.
In the same medium pot over medium-high heat, add the ground pork, sea salt, and black pepper, browning the meat until it is fully cooked through.
Add the minced garlic, grated ginger, and sliced shiitake mushrooms to the pork, sautéing for about 2 minutes until the aromatics are fragrant.
Pour in the chicken bone broth and tamari, bringing the mixture to a gentle simmer.
Add the baby bok choy to the broth and cook for 2 minutes until the leaves are wilted but the stems remain crisp.
Place the cooked soba noodles in a deep bowl, pour the pork and broth mixture over them, and top with the peeled and halved soft-boiled egg.
Drizzle with toasted sesame oil and garnish with thinly sliced green onions before serving warm.