Heat a medium saucepan over medium-high. When hot, add 1 tablespoon oil. Add the ginger and white parts of the scallions and cook for 1-2 minutes on medium heat until fragrant and the ginger and scallions have softened. Add the mushrooms and cook for 4-5 minutes, stirring occasionally, until they are lightly golden. Add star anise and vegetable stock, cover and cook for 15 minutes.
Prepare the noodles: Bring a large pot of salted water to a boil. Add the rice vermicelli and cook according to package instructions, about 3 minutes, until the noodles are just tender. Drain, rinse with cold water and drain well again. Separate into 4 serving bowls.
Place the green parts of the scallion, ginger, chili and salt into a heatproof bowl. Place the oil in a small saucepan, and heat over medium until it bubbles, about 1-2 minutes. Remove from the heat and very carefully pour the oil over the scallions, ginger and chili.
Add the Asian greens to the broth and cook for just 30 seconds until bright green. Season the broth with kosher salt and white pepper; discard the star anise.
To serve, ladle hot broth and greens over each bowl of noodles, and finish each with the spicy scallion and ginger topping.