Soak the dangmyeon in warm, not hot, water for 30 minutes, until a strand bends without snapping.
Combine the sliced beef with the marinade ingredients and set it aside for at least 10 minutes while you prep the vegetables.
Bring a pot of water to a boil. Drain the soaked noodles and boil them for 5 to 6 minutes, until they've turned fully translucent and taste chewy rather than starchy.
Drain the noodles, rinse briefly under cold water, and toss immediately with 1 tablespoon of the sesame oil so they don't clump together while everything else cooks.
Wipe out a wide pan or wok. Cook the spinach alone over medium heat for about 1 minute, just until wilted, season with a pinch of salt, and move it to its own bowl.
In the same pan, stir-fry the carrots alone for 2 to 3 minutes, until just tender with a little bite left, then move them to their own bowl.
Stir-fry the onion alone for 2 minutes until soft and starting to turn gold, then the bell pepper alone for about 1 minute, keeping each one in a separate bowl.
Stir-fry the mushrooms alone for 3 minutes, until they've released their liquid and started to catch a little color at the edges.
Cook the marinated beef alone over medium-high heat for 3 to 4 minutes, until just cooked through and slightly caramelized at the edges.
If using the egg garnish, fry the whites and yolks separately into thin sheets, let them cool flat, then slice into fine strips.
Return the noodles to the pan or a large mixing bowl with the remaining soy sauce, sugar, and 1 tablespoon of sesame oil. Toss over low heat for 1 minute so the noodles absorb the sauce.
Add the beef and every vegetable back in and toss everything together for 1 minute, just until combined and warmed through. Finish with sesame seeds and the egg strips.