In a medium bowl, whisk together the miso, vinegar, sesame oil and ginger. Add the chicken breasts and spoon the marinade all over to coat. Cover with plastic wrap and transfer to the refrigerator to marinate for at least 1 hour or up to 3 hours.
Heat a skillet over medium-high heat and add the chicken breasts. Cook, without disturbing, until golden brown on one side, about 4 minutes. Flip and continue cooking until the chicken is cooked through, about 4 minutes longer. (If your chicken breasts are large, and you find that the outside is getting too browned before the breasts are cooked through, you can transfer the chicken to a 350˚preheated oven to finish cooking).
Cook the rice noodles according to the package directions and set aside.
In a medium stockpot, add the chicken stock, soy sauce and ginger and bring to a gentle simmer over moderate heat. Reduce the heat to low so that the soup is no longer simmering, but is kept warm. Add the cooked rice noodles and mushrooms and leave over low heat for 5 minutes.
To serve, divide the broth, noodles, and mushrooms between four bowls. Slice the chicken breasts into thin (¼-inch) slices and arrange over top of each. Garnish with a few sprigs of cilantro and thinly sliced radish. Serve sriracha on the side, if desired.
Shitake mushrooms would work very well in this recipe too!