Bring a large pot of well-salted water to a boil. Cook the pasta until al dente according to the package directions.
Before draining, reserve 1 cup of the pasta cooking water. Drain the pasta and set aside.
While the pasta cooks, heat the olive oil in a large skillet over medium heat. Add the minced garlic and cook for about 30 seconds, just until fragrant.
Add the tomato paste and cook for 5–7 minutes, stirring often, until it darkens slightly and turns thick and glossy.
Reduce heat to low and stir in the heavy cream. Simmer for 1–2 minutes until smooth and combined.
Season with salt, black pepper, and red pepper flakes (if using).
Add the drained pasta to the skillet and toss to coat. Add reserved pasta water a little at a time, stirring, until the sauce becomes glossy and clings to the pasta.
Stir in the Parmesan until melted. Taste and adjust salt and pepper as needed.
Serve immediately topped with fresh basil and extra Parmesan.