Cook the Pasta: Bring a large pot of salted water to a boil and add the pasta. Cook until al dente, reserving 1 cup of the pasta water while the pasta is boiling. Drain and set aside.

Prep: In the meantime, combine the rice vinegar and cane sugar together in a small bowl. Mix well, until the sugar is dissolved. Set aside.

Aromatics: Melt 2 tablespoons of butter in a large saucepan over medium heat. Add the shallot and sauté for 3 to 5 minutes, until translucent and golden. Add the garlic and sauté for 1 minute.

Make the Sauce: Add the gochujang, tomato paste, and gochugaru (if using) to the aromatics, using a spatula to break up the paste so it is evenly distributed. Sauté for 1 minute, stirring frequently. Deglaze the pan with the rice vinegar mixture and mix well. Once the liquid from the vinegar has evaporated, add in ½ cup of the reserved pasta water. Bring to a simmer, stirring occasionally, and cook down for 3 to 5 minutes, or until you can drag a spatula through the sauce and leave a clear line.

Combine: Reduce the heat to low and add in the remaining butter, parmesan, and drained pasta. Mix well, until the cheese has melted and a thick sauce forms. If the sauce is too thick, thin it out with ~2 tablespoon increments of reserved pasta water until it reaches your desired consistency.

Serve: Transfer the pasta to a serving dish and top with finely chopped chives and extra parmesan, if desired. Serve warm; leftovers can be stored in the refrigerator for up to 5 days.