Bring a large pot of water to a boil. Add a generous pinch of salt so the water tastes lightly seasoned. Add the orecchiette pasta and cook according to package instructions until just al dente.
About three minutes before the pasta finishes cooking, add the broccoli florets directly into the same pot with the pasta. This allows the broccoli to cook quickly while staying bright green and slightly crisp.
Before draining, reserve about ½ cup of the pasta cooking water. Drain the pasta and broccoli together and set aside.
While the pasta cooks, heat 2 tablespoons olive oil in a large skillet over medium heat. Add the chickpeas in a single layer. Let them cook undisturbed for 3 to 4 minutes so they develop a light golden exterior.
Stir the chickpeas occasionally and cook for another 4 minutes until they become slightly crisp on the outside while remaining tender inside. Sprinkle with a small pinch of salt and set them aside in the skillet.
Reduce the heat slightly and add the remaining 3 tablespoons olive oil to the skillet. Add the thinly sliced garlic and cook gently for about 1 to 2 minutes until the slices turn lightly golden and fragrant. Stir frequently so the garlic does not burn.
Add the crushed red pepper flakes and black pepper to the garlic oil and stir for about 20 seconds to release their aroma.
Add the drained pasta and broccoli to the skillet. Toss gently so the pasta gets coated with the garlic olive oil.
Add the chickpeas back into the skillet and mix everything together carefully so the broccoli stays intact.
If the pasta looks slightly dry, add 1 to 2 tablespoons of the reserved pasta water. This helps create a light glossy sauce that clings to the pasta.
Sprinkle in the grated Parmesan cheese and half of the chopped parsley. Toss again so the cheese melts slightly and blends into the olive oil.
Taste and adjust seasoning with additional salt or pepper if needed.