Bring a pot of water to a boil and cook udon noodles according to package instructions. Reserve 1/2 cup of noodle water, drain the noodles, rinse with cold water, and set aside.

In a skillet over medium heat, melt butter. Add minced garlic and chopped shallots; sauté until fragrant, about 2–3 minutes.

Stir in the gochujang and cook for 30 seconds to release its flavor.

Slowly add heavy cream and reserved noodle water. Stir in soy sauce and gochugaru. Let the sauce simmer for 2–3 minutes until slightly thickened.

Add the udon noodles to the pan and toss to coat evenly in the sauce.

Plate the noodles and top with grated parmesan cheese, green onion, and an egg yolk if desired.