Heat the sesame oil in a large pot over medium heat. Add the garlic and ginger and cook for one minute until fragrant.
Add the sliced shiitake mushrooms and carrot rounds. Cook for 4 to 5 minutes, stirring occasionally, until the mushrooms soften.
Pour in the vegetable broth and bring the mixture to a gentle simmer. Cook for 8 minutes.
While the broth simmers, cook the rice noodles according to package directions in a separate pot. Drain and set aside.
Remove the pot from the heat. Ladle about half a cup of broth into a small bowl, whisk in the miso paste until smooth, then stir the mixture back into the soup.
Add the chopped bok choy and let it wilt for 2 minutes.
Divide the cooked noodles among four bowls. Ladle the hot soup over the noodles and top with sliced green onions.