Boil a large pot of water. Stir in kosher salt, then cook rigatoni until very al dente. Reserve pasta water, then drain.
Warm olive oil in a wide skillet over medium heat. Soften onion for 3 to 4 minutes, stirring.
Add garlic and cook 30 seconds until fragrant.
Stir in tomato paste and Calabrian chili paste. Toast 1 to 2 minutes until deepened in color.
Pour in crushed tomatoes. Simmer 8 to 10 minutes, stirring, then season with black pepper.
Add drained rigatoni, butter, and a splash of reserved pasta water. Toss over medium heat until glossy and coated, adding more water as needed.
Remove from heat. Fold in basil and half the Parmesan.
Divide into bowls. Tear burrata over the pasta and finish with remaining Parmesan. Serve right away.