In a large pot, combine the vegetable broth, green onion, grated ginger, and salt. Bring to a full boil, then reduce the heat and simmer for 15 minutes.
While the broth is cooking, melt the butter in a large skillet over medium heat. Add the mushrooms and sauté for about 6 minutes, or until tender, stirring frequently. Stir in the hoisin and sesame oil and cook until the sauce thickens and coats the mushrooms, about 1 minute more. Remove from heat.
Divide the rice noodles between four to six large bowls, then fill each bowl with the ginger broth. Add bean sprouts, sliced jalapeños, shiitake mushrooms, fresh basil, and cilantro and serve with lime wedges, hoisin, and chili garlic sauce.