Bring a large pot of water to a boil.
Meanwhile, season both sides of the chicken with salt and pepper.
Heat the oil in a large pot set over medium-high heat. Add the chicken and sear on both sides until golden brown, about 3 minutes per side. Your chicken doesn't have to be completely cooked through, as it will continue cooking in the soup later. Transfer to a plate and set aside.
Reduce the heat to medium and add the onions, jalapeño (if using), and garlic. Sauté, stirring, until onions are tender, about 2 minutes. Add the mushrooms and continue to cook until softened, about 2 minutes.
Add the ginger, miso, coconut aminos, and 1/2 cup of broth. Cook, stirring, for 1 to 2 minutes, until the miso is well incorporated.
Slowly pour in the remaining 5½ cups of broth, stirring constantly. Bring the broth to a boil, then reduce to a gentle simmer over medium-low heat. Nestle the chicken and any of its juices in the pot. Cover and cook, simmering lightly, until the chicken is fully cooked and tender, about 10 minutes.
Using tongs, transfer the chicken to a cutting board and shred the chicken or slice it into thin strips. Return the chicken to the soup.
Stir in the green onions and sliced seaweed. Taste and adjust seasonings, if needed. Remove from heat and cover to keep warm.
Meanwhile, cook the rice noodles in the boiling water according to package instructions.
Drain the noodles and rinse under cold water to stop the cooking. To serve, divide the noodles amongst four bowls, then ladle the soup over the top. Drizzle with chili oil or sriracha, if desired.