Crack the egg into a bowl and beat well with the sugar and salt. Lightly oil a frying pan with the cooking oil. Pour in the beaten egg and spread it into a thin layer. Cook over medium-low heat until set on both sides, then remove and let cool.
Slice the cooled egg, ham, and cucumber into thin strips.
In a small bowl, combine the vinegar, cold water, soy sauce, sesame oil, and sugar. Mix until the sugar dissolves to make the hiyashi chuka sauce.
Bring a pot of water to a boil and cook the Chinese alkaline noodles (chuka noodles) according to the package directions. Drain, transfer to ice water until completely chilled, then drain thoroughly.
Arrange the chilled noodles on a plate and top with the egg, ham, cucumber, and, if desired, pickled red ginger. Pour the sauce over the top and serve immediately.
