How to Make It
In a medium bowl, toss the chickpeas with the lemon juice and 1/2 cup of the olive oil. Season with salt and pepper.
In a large pot of boiling salted water, cook the broccoli until crisp-tender, about 4 minutes. Using a slotted spoon, transfer the broccoli to a colander and rinse under cold water until cool. Add the rigatoni to the boiling water and cook until al dente.
Meanwhile, in a large, deep skillet, heat the remaining 1/4 cup of olive oil. Add the garlic and crushed red pepper and cook over moderate heat until the garlic is golden, about 3 minutes. Add the broccoli and cook until tender, about 5 minutes. Add the chickpea mixture and cook until warmed through, about 1 minute.
Drain the rigatoni, reserving 1/4 cup of the cooking water. Add the pasta to the broccoli and chickpeas along with the reserved cooking water and season with salt and pepper. Cook over moderate heat, stirring, until the rigatoni is coated with sauce. Remove from the heat and stir in 1/2 cup of the Parmesan cheese. Transfer the pasta to a bowl, sprinkle with the remaining Parmesan and serve.