To make the noodle sauce: In a medium bowl, whisk together the soy sauce, sesame oil, sugar, sesame seeds, garlic, pepper, and tablespoon of water until well combined. Set aside.
To marinate the rib eye: In a medium bowl, whisk together the soy sauce, mirin, garlic, sugar, and sesame seeds until well combined. Add the sliced rib eye. Mix well to ensure that the rib eye is completely coated. Cover and refrigerate for at least 15 minutes.
To make the japchae: In a large skillet, heat 1/2 teaspoon of the sesame oil over medium heat, then add the spinach and garlic. Cook, stirring occasionally, until the spinach is wilted, 2 minutes. Remove the from the heat and stir in the pepper, 1/2 teaspoon of the sesame seeds, and the salt. Transfer to a plate and set aside. Wipe the pan clean.
In a medium saucepan, bring 5 cups of water to a boil. Add the noodles and cook, stirring frequently to prevent clumping, until tender, 10 to 15 minutes.
Meanwhile, in the wiped out skillet, heat 1 tablespoon of the sesame oil over medium heat. Add the onion and cook, stirring occasionally, until tender, 3 to 5 minutes. Transfer to the plate with the spinach and set aside.
In the same pan, heat another tablespoon of sesame oil over medium heat. Add the carrots and mushrooms and cook until tender and soft, 5 to 7 minutes. Transfer to the plate with the onion and set aside.
Drain but do not rinse the noodles. Add them to the pan along with another tablespoon of the sesame oil and toss. Transfer the noodles to the bowl with the noodle sauce and mix together.
In the same pan, heat the remaining 1 tablespoon sesame oil over high heat. Add the marinated rib eye with its marinade and cook, stirring often, until there is less than 1 tablespoon of liquid left in the pan, 4 to 6 minutes.
Transfer the vegetables and cooked rib eye to the bowl of noodles and mix together, using tongs.
Transfer to a serving dish and sprinkle with the remaining 1 teaspoon sesame seeds. Japchae can be served hot, room temperature, or cold.
Store in an airtight container in the refrigerator for up to 3 days.