Bring 12 cups of water to a boil in a large pot. Add 1 tablespoon salt. Cook pasta according to package directions until al dente. Reserve 1 cup of pasta cooking water before draining.
Heat olive oil in a large nonstick pan over medium-high heat. Add onion and garlic, cooking until softened and golden, 3–5 minutes. Stir in tomato paste and cook for 2 minutes, stirring constantly, until it slightly darkens in color (this adds flavor!). Lower heat if needed.
Stir in cream, chicken stock, red pepper flakes (if using), and salt and pepper to taste (I add 3/4 teaspoon salt and 1/4 teaspoon pepper). Simmer, stirring often, until thickened, 3–4 minutes. Add in Parmesan, stirring constantly, until melted.
Add pasta and 1/2 cup reserved cooking water to the sauce. Toss over medium heat for 2 minutes, until pasta is well-coated. Add more cooking water, 1/4 cup at a time, if needed to loosen the sauce.
Plate immediately and top with extra Parmesan.
Video
Notes
Note 1: Grab a block of Parmesan and then grate using a microplane. Grate until fluffy and loosely fill 1 full cup. Avoid pre-grated Parmesan, which makes the sauce grainy.Storage: Best enjoyed right off the stovetop. Store leftovers in an airtight container in the fridge for 3–4 days. Reheat gently with a splash of cream or pasta water to loosen the sauce.