Start by making the sauce. In a mixing bowl, combine the hot water and sugar. Whisk until the sugar dissolves completely.
Add in the soy sauce, vinegar, sesame oil, and chili oil if you desire. Whisk to combine.
Cover the bowl and refrigerate until serving time.
Next, make the thin egg crepe. Whisk the eggs, sugar, and water together until well combined.
In a large nonstick frying pan, heat the vegetable oil over high heat until shimmering.
Pour in the egg mixture and quickly turn the pan to coat the bottom. Cook for 1-2 minutes until the top is set.
Once the top of the egg is set, flip it over and cook the other side for another 1 minute or so until cooked. Remove from heat and set aside to cool. Pat a paper towel on it to absorb any extra oil.
While the egg crepe is cooling, diagonally slice the cucumber so you have elongated oval slices. Then, finely julienne those oval slices to get thin strips. Set side.
Lay the ham slices so they are stacked together. Finely julienne the ham into thin strips.
Thinly slice the egg crepe into julienne slices.
Bring a pot of water to a boil. Add the hiyashi chuka noodles and separate them in the water using chopticks or tongs. Cook for 2-3 minutes until al dente. (I normally will take them off the heat in the last minute- they cook very fast!)
Drain the noodles in a colander and immediately rinse with cold running water. (Tip: I like to prepare a bowl of ice-water to dip my rinsed noodles in- totally optional but it chills them faster!)
Divide the drained noodles among serving dishes. Arrange the toppings onto each plate and garnish with the pickled ginger, if desired. Serve with the chilled dressing on the side. Enjoy!