Cook the soba noodles according to package directions. Drain and set aside.
Add 8 cups water to medium pot (a Dutch oven works well), along with the garlic, ginger, white parts of the scallions and the salt. Bring to a boil over high heat and then reduce to a simmer. Add the chicken. Poach for about 15 minutes, depending on the thickness of the chicken breasts you might need to go a few minutes longer than that. Remove the chicken and place on a plate.
Use a slotted spoon to remove the garlic, ginger, and scallions from the broth. Discard, then add the soy sauce and mirin to the pot. Taste and adjust the seasoning. Add the bok choy and mushrooms and simmer for a few minutes until tender. Shred the chicken and return it to the pot.
If the noodles are sticky, quickly rinse them under warm water and divide between 4 bowls (you may have more noodles than you need). Ladle the broth over the noodles, dividing the chicken, mushrooms, and greens evenly between the bowls. Add a drizzle of sesame oil to each bowl and serve with hot sauce on the side.