Boil a pot of water. Cook the Udon Noodles for 1 minute until the noodles loosen up. Drain and transfer to a bowl. Add in Sesame Oil. Mix well and set aside.
In a wok, on medium-high heat. Add the Butter. Once almost melted, add in the All-purpose Flour. Whisk well. Then add in some Chili Flakes and the Whole Milk. Keep whisking until a nice sauce is formed. Turn the heat down to low.
Next, add in Gochujang, Parmesan Cheese and half of shredded Mozzarella cheese. Stir well to combine. And the sauce looks thick and cheesy.
Add in the Udon Noodles and the remaining Mozzarella Cheese. Combine everything well.
Transfer to a serving bowl. Sprinkle over some Green Onions and Gochugaru. Ready to enjoy!