Cook the noodles: Bring a large pot of water to a boil. Add the somyeon noodles and cook according to package directions (usually 3-4 minutes). Drain immediately.
Rinse the noodles: Run the cooked noodles under cold water for 1-2 minutes, tossing them with your hands to cool them down completely and remove excess starch. Drain well and set aside.
Make the sauce: In a small bowl, whisk together the gochujang, sugar, rice vinegar, soy sauce, sesame oil, minced garlic, and gochugaru (if using). Taste and adjust sweetness or heat to your preference.
Prep the vegetables: While the noodles are cooling, julienne the cucumber and carrot into thin matchsticks. Shred the lettuce and prepare your soft-boiled eggs.
Assemble the dish: Place the cold noodles in a large bowl. Add a handful of ice cubes to keep them extra cold. Pour the sauce over the noodles and toss well until every strand is coated.
Add toppings: Arrange the julienned vegetables on top of the noodles. Add the soft-boiled egg (cut in half), sprinkle with sesame seeds, and add nori strips and green onions if desired.
Serve: Mix everything together right before eating so you get a bit of noodles, sauce, and veggies in every bite. Enjoy immediately while cold!