Bring a large pot of lightly salted water to a boil. Cook spaghetti in the boiling water, stirring occasionally, until tender yet firm to the bite, about 12 minutes.
Meanwhile, combine ice water, vinegar, soy sauce, sugar, and sambal oelek in a bowl. Refrigerate sauce to chill until spaghetti is cooked.
Rinse cooked spaghetti in cold water until totally cooled off; drain well.
Divide sauce between 2 chilled servings bowls. Add equal amounts of spaghetti, cucumber, kimchi, peanuts, egg, and ham to each bowl. Serve immediately.
