Prepare Onsen Tamago. Boil water in a heavy pot. Remove the pot from the heat and carefully add the eggs. Cover the pot and let the eggs sit in the hot water for 15 minutes. After 15 minutes, remove the eggs from the hot water and set them aside until ready to use. Prepare seasoned beef. Cut the sliced beef into large pieces. Slice the onion thinly and cut the ginger into thin matchsticks. Heat oil in a frying pan over medium heat, and add the beef. Once the beef changes color, add the ginger and onion, and cook until soften. Then, add sugar, soy sauce, mirin, sake, and water. Cook for 2-3 minutes. Allow the mixture to cool in the pan.
Prepare Udon. Cut the green onions thinly and diagonally, then set them aside. Heat the dashi until it boils. Add salt, soy sauce, mirin, and sake to the dashi. Next, add the udon noodles to the soup and let them simmer for 2-3 minutes. If the udon noodles are frozen, cook them for a couple of minutes longer.
Assemble. Divide the noodles and soup evenly into bowls and top each bowl with the beef and sliced green onions. Crack an egg into each bowl. If desired, sprinkle with shichimi to add flavor.