Bring a pot of water to a rolling boil and cook the noodles according to the package time, usually 3 to 4 minutes.
While the noodles cook, whisk together the gochujang, gochugaru, vinegar, sugar, soy sauce, garlic, and sesame oil in a large bowl.
The moment the noodles are done, drain them and rinse immediately under cold running water, rubbing them gently between your hands, until they're fully cooled and no longer starchy to the touch.
Drain well and add the noodles to the bowl with the sauce.
Toss thoroughly with tongs or chopsticks until every strand is coated.
Divide between two bowls and top with cucumber, kimchi if using, the egg, and a scatter of sesame seeds.
Serve immediately.