Bring a large pot of salted water to a boil. Add the pasta and cook until al dente, about 1 minute less than package directions. Reserve ½ cup of pasta water, then drain and toss with olive oil.

In the same pot, melt the butter over medium heat. Add minced garlic and cook until fragrant, about 30 seconds to 1 minute.

Stir in gochujang, toasting for 1-2 minutes until it deepens in color.

Pour in heavy cream and half of the reserved pasta water. Stir until smooth, then season with salt, black pepper, and smoked paprika. Bring to a simmer.

Add cooked pasta and grated parmesan to the sauce. Stir gently and cook on low for 1-2 minutes until well-coated.

Adjust seasoning as needed and serve immediately, topped with additional parmesan.