Boil somyeon according to package instructions. Rinse well under cold water to stop the cooking and remove excess starch. Drain thoroughly.
Air fry eggs at 300°F (150°C) for about 12 minutes. Transfer immediately to an ice bath for easy peeling.
In a large mixing bowl, mix together gochujang, rice vinegar, soy sauce, sugar, kimchi juice, sesame oil, water, garlic, and gochugaru until smooth.
Add the cooled noodles to the sauce and toss until evenly coated.
Chop the kimchi (a trick: use kitchen scissors inside a glass to avoid staining your cutting board). Slice cucumbers thinly.
Add the sauced noodles to a bowl. Top with chopped kimchi, cucumber slices, halved soft-boiled eggs, and a sprinkle of sesame seeds.