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In large pot, bring water to rapid boil.
Add broccoli; bring water back to boil for 2 minutes. Do not drain water! Using slotted spoon, immediately transfer broccoli to colander and place under cold running water to stop further cooking.
Bring water back to boil, add Cavatelli, and cook according to package (reserving 1 cup pasta cooking water before draining).
In large blender or food processor, blend broccoli, basil, garlic, nuts, olive oil, salt, and Parmigiano until smooth. Slowly pour in 1 cup reserved pasta water. If sauce is too thick, add more pasta water one tablespoon at a time.
Toss hot pasta with sauce until completely coated. Top with extra Parmesan and chili flakes, if desired.