Boil pasta in salted boiling water until al dente. When ready, save 1/2 cup of pasta water and drain the rest. Drizzle with olive oil to prevent burning.
Melt the butter over medium heat. Once it starts to sizzle, add the minced garlic and cook for 30 seconds to 1 minute, stirring very often, or until fragrant.
Then, add the heavy cream and half of the reserved pasta water. Stir until the sauce is creamy and smooth. Add salt and black pepper, and smoked paprika.
Toasting the gochujang paste in oil or butter caramelizes it, and gives it a nutty flavor. It's an essential step to add flavor and sweetness to the sauce.