Bring a small pot of water to a boil, carefully lower the egg into the water, and cook for 6 minutes and 30 seconds. Immediately transfer to an ice bath, peel when cool, and set aside.
In a medium pot, heat the toasted sesame oil over medium heat. Add the minced garlic, grated ginger, and sliced shiitake mushrooms, sautéing for 3 minutes until fragrant.
Pour in the chicken broth, tamari, and sriracha sauce. Bring the liquid to a gentle simmer.
Add the whole chicken breast to the simmering broth. Cover and poach for 10-12 minutes or until the internal temperature reaches 165°F. Remove the chicken, let it rest for 5 minutes, then slice into thin strips.
While the chicken rests, add the brown rice ramen noodles to the simmering broth and cook according to package directions (usually 3-4 minutes).
During the last minute of noodle cooking, stir in the chopped baby bok choy until just wilted and bright green.
Pour the noodles and broth into a large bowl. Top with the sliced chicken, the soft-boiled egg halved lengthwise, and a garnish of sliced scallions.