Boil the noodles in salted water until just tender. Reserve a splash of cooking water, then drain.
Whisk gochujang, soy sauce, brown sugar, rice vinegar, and sesame oil in a small bowl.
Melt butter in a medium skillet over medium heat. Add garlic and cook 30 seconds until fragrant.
Stir in corn, mayonnaise, milk, salt, pepper, and gochugaru. Simmer 2–3 minutes until creamy.
Scatter mozzarella over the corn mixture. Cover until melted and bubbly, 2–3 minutes (or broil briefly).
Toss hot noodles with the sauce. Add a spoonful of cooking water if needed to coat.
Pile noodles into bowls and spoon the cheesy corn on top. Finish with green onions and serve hot.