Prepare the pesto: In a blender or food processor, combine the basil leaves, garlic, and pine nuts. Blend until finely chopped. Slowly drizzle in the olive oil while blending until the mixture forms a smooth paste. Add the Parmigiano-Reggiano, Pecorino, and salt, and blend again. Taste and adjust seasoning if needed.
Cook the fusilli: Bring a large pot of salted water to a boil. Add the fusilli and cook according to the package instructions until al dente, usually about 10-12 minutes. Reserve a cup of pasta water, then drain the pasta.
Combine pasta and pesto: In a large bowl, toss the hot fusilli with the pesto sauce, adding a little reserved pasta water to loosen the sauce and help it coat the pasta evenly.
Serve: Serve immediately with an extra sprinkle of Parmigiano-Reggiano on top.