Puree the tomatoes using an immersion blender or food processor. Set aside.
In a saucepot over medium heat, combine the olive oil, pepper flakes and garlic. Gently toast the garlic until golden brown, being careful not to burn it. Add the tomatoes and bring to a simmer; simmer 5 to 8 minutes.
Bring a large pot of salted water to a boil. Add the rigatoni and cook for 2 minutes less than the package instructions. Reserve about 1/4 cup of the pasta water, then drain the pasta.
Add the pasta to the pot with the sauce. Add 2 tablespoons of the reserved pasta water and cook until thickened, about 3 minutes. Add the butter, half of both cheeses and half of the basil and cook, tossing frequently, until the pasta is nicely coated. Season with salt if necessary.
Transfer to a serving bowl, top with the remaining cheese and basil and drizzle with olive oil.