Put the beef in a medium bowl with 1 tablespoon sesame oil, 2 tablespoons soy sauce, 1 tablespoon brown sugar and 1 tablespoon minced garlic and mix until the meat is completely coated. Set aside.
Put the shiitake mushrooms in a small bowl and pour in enough hot water to cover. Set aside to soften for 25 minutes. Remove the softened mushrooms from the liquid, slice thinly and set aside.
Heat 2 tablespoons sesame oil in a large skillet. Add 2 tablespoons minced garlic, the shiitake mushrooms, shishito peppers, carrots, onions and red bell peppers and cook until slightly soft but still crunchy, for 3 minutes. Add 2 tablespoons soy sauce and continue to cook for an additional minute. Transfer to a bowl and set aside. Set the skillet aside for cooking the beef.
Prepare a medium bowl of ice water. Bring a medium saucepan of salted water to a boil, and then add the spinach and cook for 1 minute. Remove with a slotted spoon and immediately add to the ice water. Drain the spinach and squeeze with your hands to remove the excess liquid. Roughly chop and set aside.
In the same pan of boiling water, add the noodles and cook according to the package directions. Drain the noodles in a strainer and rinse thoroughly with cold water. Add 2 tablespoons sesame oil to the noodles and toss with your hands or tongs so they are well coated and do not stick together. Leave the noodles in the strainer and set aside.
Place the reserved skillet over medium-high heat and cook the beef with the marinade until brown, for 3 to 4 minutes.
Combine the noodles, vegetables and beef in a large bowl. Add the remaining 7 tablespoons sesame oil, 7 tablespoons soy sauce, 1/4 cup brown sugar, the remaining garlic, the mirin, fish sauce, rice vinegar and black pepper. Mix together, using your hands or tongs, until all the ingredients are incorporated.
Transfer to a large serving bowl and sprinkle with sesame seeds.