Bring a large pot of heavily salted water to a boil. Add the pasta and cook according to package instructions until tender. Before draining, reserve 1/2 cup of the pasta water. Drain and set aside.
In a large skillet over medium heat, add the butter and olive oil together. Once melted and warm, add the finely diced shallot and cook until soft and translucent, about 3–4 minutes.
Stir in the minced garlic and cook for about 1 minute until fragrant. Add the finely chopped Calabrian chilies and sauté briefly to release their flavor.
Add the tomato paste and cook, stirring constantly, for 4–5 minutes until it darkens and becomes slightly caramelized.
Stir in the crushed San Marzano tomatoes along with the reserved 1/2 cup pasta water. Let simmer for 3–4 minutes to combine.
Lower the heat and stir in the heavy cream. Add the dried basil, salt, and black pepper. Simmer gently until smooth and slightly thickened.
Using an immersion blender, blend the sauce directly in the skillet until completely smooth. Alternatively, carefully transfer the sauce to a blender, blend until silky, and return it to the pan over low heat.
Stir in the freshly grated Parmesan until fully melted into the sauce. Taste and adjust with more salt, black pepper, and red pepper flakes depending on your preferred spice level.
Add the cooked pasta to the sauce and toss until glossy and fully coated.