Bring a large pot of well-salted water to a boil and cook the noodles until tender, about 4 minutes. Drain well, transfer to a large bowl, toss with the sesame oil, cover, and keep warm.
Meanwhile, combine the soy sauce, mirin, sugar, ginger, garlic, and 1 cup water in a 10-inch straight sided sauté pan. Bring to a boil, and then add the steaks. Turn the heat down and simmer gently, flipping once, until medium rare (130°F), 12 to 16 minutes. Transfer the steaks to a cutting board, reserving the liquid in the pan. Discard the garlic and ginger.
Return the liquid to a boil. Add the zucchini and edamame, return to a boil, and then lower to a simmer. Cook until the vegetables are just tender, about 2 minutes. Using a slotted spoon, transfer the vegetables to the bowl of noodles. Toss well, cover, and keep warm.
Boil the cooking liquid until reduced by half, about 5 minutes.
Thinly slice the steaks across the grain. Arrange the noodle mixture on a platter or divide it among 6 shallow bowls. Top with the beef and the pickled ginger, if using. Drizzle some of the sauce over the beef and garnish with the scallions and sesame seeds, if using. Serve, passing the rest of the sauce at the table.