Bring a large pot of water to a boil. Add noodles, and cook until al dente according to package instructions, about 8 minutes. Drain in a colander, toss with sesame oil, and return to pot. Keep warm.
Meanwhile, in a medium saute pan, heat the vegetable oil over medium heat. Add ginger, shallots, and mushrooms; cook, stirring constantly, until the mixture begins to soften and turn golden brown, about 2 minutes.
Stir chicken stock, vinegar, and soy sauce into pan, and bring to a simmer. Cook until mushrooms are very tender, about 5 minutes. Add spinach and scallions, and stir to combine. To serve, divide noodles among four shallow bowls; ladle soup over noodles.