Bring a pot of water to a boil. Add the pasta and cook 7–8 minutes. Drain, reserving some pasta water.
Add the butter to a saucepan. Melt over low heat, then increase to medium. Stir until brown bits appear and the butter turns golden.
Add the chili crisp oil, shallot/onion, sun-dried tomatoes, and garlic. Cook 3–4 minutes until softened.
Add the hot pepper paste and tomato paste. Cook 2–3 minutes until darkened. Add a splash of vodka (optional) and cook 2–3 minutes more.
Add the heavy cream, milk, and basil. Stir well and cook 1 minute.
Season with salt and chili flakes. Stir in the grated cheeses and 1–2 tbsp pasta water for a smooth sauce. Add the cooked pasta, toss to combine, and serve.