Soak the dried shiitakes in a bowl. Meanwhile, mince the beef, and make the beef marinade. To make the marinade, add the soy sauce, sugar, sliced scallions, minced garlic, sesame seed salt, sesame oil, and crushed black pepper in a bowl. Add the minced beef and set aside.
Take the cucumber and slice in half, lengthwise. Then, slice diagonally into 1/8" thick pieces. Place in a bowl and add 2 spoonfuls of seat salt. Mix and let sit to allow the cucumbers to sweat.
For the eggs, crack and separate the white and the yolk. Briefly beat each with a fork and add a pinch of salt. Set a small skillet on medium heat and add a slight drizzle of cooking oil. Pour the yolk into the skillet and fry to be thin and round. Remove to let cool, then fry the white in the same way. Set the fried eggs to the side and let cool.
Once the shiitakes are soft and moistened, squeeze out the water and slice into thin pieces. Add to the beef mixture and set aside.
Make the marinade for the noodles by mixing together all the ingredients listed above in a bowl.
Bring a large pot of water to a boil. Meanwhile, 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, making sure they do not burn.
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.
Take the cooled fried eggs and slice in half, to semi-rounds. Then, slice across 1/8" thick, so they look like flat noodles. Set aside.
Now the water should be boiling. Add the noodles 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.