In a large soup pot, heat 1 tbsp of oil over medium high heat. Pat the chicken breast dry and season with salt and pepper on both sides. Add the chicken to the pot and cook, undisturbed, for 8 minutes. Flip, add the shallots, and cook for 4 minutes, and then lower the heat and cover, cooking for another 5 minutes, or until the centre of the chicken is 165 F. Remove the chicken and browned shallots to a cutting board and tent with foil. Place the pot back over medium heat. Add the remaining oil, mushrooms and a splash of broth. Cook, stirring often, until mushrooms are softened and just starting to brown.
Add the remaining broth, ginger, garlic, soy sauce, rice vinegar, sugar, and red pepper flakes to the pot. Bring to a boil, and then add the ramen noodles, carrots and pak choy.
Reduce the heat and cover, maintaining a gentle simmer until the ramen has softened and the vegetables are tender-crisp, about 2 minutes.
Turn off the heat and ladle a cup of broth into a measuring cup. Whisk in the miso paste and pour it back into the pot.
Slice the chicken. Divide the soup between 4 bowls, and top with chicken, shallots, and green onion.