Bring a large pot of water to a boil. Add the frozen udon noodles and cook according to package instructions, about 5-7 minutes. Reserve ½ cup noodle water, then drain and rinse noodles under cold water.

In a large skillet, melt 2 tablespoons of butter over medium heat. Add minced garlic and chopped shallot, sauté for 2-3 minutes until fragrant and translucent.

Stir in gochujang and cook for 30 seconds. Gradually pour in heavy cream and reserved noodle water, stirring constantly. Add soy sauce and gochugaru, simmer for 3-5 minutes until thickened.

Add the cooked udon noodles to the skillet, tossing to coat evenly. Cook for 1-2 minutes over low heat to absorb the sauce flavors.

Transfer noodles to a bowl. Garnish with Parmesan cheese and green onions. Serve immediately while hot.