In a large (at least 8-quart) pot, bring 5 quarts water to a boil. Add the noodles, shaking them over the baking sheet to remove excess starch. Cook, stirring occasionally, until a noodle rinsed under cold water is tender, 10 to 15 minutes. Drain in a colander, rinse under cold water and drain again.

While the noodles cook, thinly slice the cucumber on the diagonal. Stack several slices and cut lengthwise into matchsticks. Repeat with the remaining slices. Set the cucumber aside. Meanwhile, in a small bowl, stir together the cornstarch and 2 tablespoons water; set aside.

In a large Dutch oven over medium-high, heat the sesame oil until shimmering. Add both ground meats and cook, stirring and breaking the meat into small pieces, until browned, 4 to 5 minutes. Add the mushrooms, bamboo shoots (if using), garlic and ginger; cook, stirring, until fragrant, 1 to 2 minutes. Stir in the broth, sake, chili-bean sauce, miso, soy sauce and mirin. Bring to a simmer, then reduce to medium and cook until the liquid has reduced by half, 6 to 7 minutes.

Stir the cornstarch slurry to recombine, then stir it into the meat-mushroom mixture. Cook, stirring constantly, until the sauce returns to a simmer and has thickened, about 1 minute. Add the noodles, toss well and cook, stirring, just until the noodles are heated through, 2 to 3 minutes. Serve topped with scallions and cucumber.