Cook Noodles: Bring a large pot of water to boil. Add noodles and cook according to package directions until just tender (usually 3-4 minutes for somyeon).
Cool Noodles: Immediately drain noodles and rinse thoroughly under cold running water until completely cooled. Drain again and set aside in refrigerator.
Make Sauce: In a mixing bowl, combine gochujang, soy sauce, rice vinegar, sesame oil, sugar, minced garlic, and grated ginger. Whisk until sugar completely dissolves and sauce is smooth. Adjust seasoning to taste.
Prepare Toppings: While noodles chill, julienne the cucumber into thin matchsticks. Slice hard-boiled eggs into rounds. Shred seaweed into thin strips. Finely chop green onions.
Assemble: Place chilled noodles in serving bowls. Pour sauce over noodles and toss thoroughly until evenly coated.
Garnish: Top each bowl with cucumber julienne, egg slices, and shredded seaweed. Sprinkle with chopped green onions and sesame seeds.
Serve: Present immediately while noodles are still cold for the best experience.