Soak the dried shiitakes in a bowl. While the shiitakes are soaking, mince beef and begin to make the beef marinade. To make the marinade, add soy sauce, sugar, sliced scallions, minced garlic, sesame seed salt, sesame oil, and crushed black pepper in a bowl. Toss in the minced beef and set aside.
Take the cucumber and slice in half, lengthwise. Then slice across the length. Place in a bowl and add 2 spoonfuls of sea salt. Mix and let sit to allow cucumbers to sweat.
Crack the eggs and separate the white and yolk. Beat each with a fork and add a pinch of salt. Set a skillet on medium heat and add a drizzle of cooking oil. Fry the yolk in the skillet. Remove, then fry the white in the same way. Set the fried eggs to the side and let cool.
Squeeze out the water from the shiitakes and slice into thin pieces. Add to beef and set aside.
Make noodle marinade by mixing together all the ingredients listed in a bowl.
Bring a large pot of water to a boil and next to it heat a large skillet to medium-high heat and add a drizzle of cooking oil. Take a thin linen towel and place in the middle a handful of sliced cucumbers. Twist the towel to squeeze out all the water. Place into the skillet and repeat with all cucumbers. Sautee the cucumbers for about 2 minutes.
Transfer the sauteed cucumbers to a bowl, and add the marinated beef to the skillet. Bring heat to medium high, and sautee for 5 minutes until the juices from the beef release then are slightly evaporated. Transfer to another bowl.
Slice the fried eggs in half, so they become half moons. Then, slice across like flat noodles. Set aside.
Add noodles to boiling water and cook until tender. Transfer to cold running water until the noodles are completely cooled. Drain the noodles and transfer to a bowl and mix with the noodle marinade.
In a large bowl, add the marinated noodles. Then layer each element: cucumber, beef mixture, and fried eggs. Garnish with a sprinkle of minced red chile pepper.