Heat several quarts of water in a large pot. When water comes to a boil, salt it lightly and add the broccoli florets. Cook broccoli for 2 minutes to blanch. (The florets should be crisp-tender and bright green.) Meanwhile, fill large bowl with ice cubes and cold water. Using a slotted spoon, transfer broccoli to ice water and allow to cool for several minutes. Drain broccoli and put aside.
Heat 3 tablespoons of olive oil in heavy large skillet over medium low heat. Add garlic pieces to oil and cook until golden brown, turning often. Do not allow garlic to burn. Remove garlic from oil with slotted spoon and discard.
Add onion to oil in skillet. Raise heat to medium. Cook for 7-8 minutes, stirring often, until beginning to wilt. Add red bell pepper and prosciutto, stirring well. Cook for an additional 5 minutes, stirring often.
Meanwhile, return water in large pot to boil. Add pasta and cook according to package directions.
When pasta has 3 minutes left to cook, add capers and olives to onion mixture and heat through.
When one minute of cooking time for pasta remains, add broccoli to onion-prosciutto mixture.
Drain pasta and return to pot. Add butter and toss until evenly coated.
Add pasta to onion-broccoli mixture and toss.
Season with black pepper to taste (we used a lot) and sprinkle with a bit of Parmesan.
Serve with more Parmesan on the side for everyone to sprinkle on according to taste.