Bring a large pot of water to a boil, salt it, add the spaghetti and cook until al dente. Drain.
While the pasta is working, in a large nonstick skillet, heat the oil over high heat until rippling. Add the chicken and stir-fry until golden, 3 to 4 minutes. Add the cabbage, scallions, mushrooms, garlic and ginger and stir-fry for 2 minutes. Season with a little salt and the pepper and keep warm.
In a large bowl, combine the tamari and sesame oil. Stir in the spaghetti and toss to coat. Sprinkle the sesame seeds on top. Using tongs, serve the pasta in bowls and top with the chicken stir-fry.