Soak the frozen udon noodles in hot water for about a minute to separate the strands. Drain and set aside.If using dried shiitake mushrooms, soak them in hot water until tender and remove the tough stems.
Make the sauce by combining soy sauce, mirin, sake, oyster sauce, and sesame oil. Set it aside.
Slice the beef thinly and across the grain. Season lightly with salt and pepper, then add a tablespoon of sauce. Mix together.
Set your wok over high heat. Add a bit of oil and brown the beef quickly for about 3 minutes. Avoid over-stirring to get a good sear.
Push the meat aside, then sear the mushrooms for about a minute.
Reduce the heat to medium. Add garlic, onions, and carrots. Sauté them for about a minute, adding more oil as needed.
Combine the cabbage and let it cook for another minute.
Add the noodles and sauce. Toss them together until the noodles are heated through, and the sauce is mostly absorbed.
Add the scallions and turn off the heat.
Garnish with more scallions, togarashi, and bonito flakes.
Notes
Udon noodles: These are thick, chewy noodles made of wheat flour. Fresh or frozen udon noodles are best; dried ones lack the same texture.
Beef: Sirloin, boneless short ribs, and flank steak are good choices. Inexpensive but less tender cuts like chuck or top round steaks with sufficient marbling (white streaks of fat) will also work. Slice them thinly across the grain to make them more flavorful and tender.
Mirin: A sweet Japanese rice wine that adds sweetness and depth. You can use "aji mirin" or "sweet cooking rice seasoning," a commercialized version you're likely to find in the US. It has a lower alcohol content and corn syrup to mimic the taste of real mirin (hon mirin). In a pinch, you can combine sake and sugar. Use water and sugar if you can't consume alcohol.
Sake: An alcoholic drink made from fermented rice that enhances flavors. You can use inexpensive sake or "cooking sake" with added salt. In a pinch, substitute with Chinese cooking wine or dry sherry. The alcohol content burns off during cooking; if you can't consume alcohol, replace it with water.
Oyster sauce: It is a thick, umami-rich brown sauce with a mild briny flavor like oysters.
Sesame oil: It has a strong nutty flavor and aroma; use the dark-colored variety made from toasted sesame seeds.