Bring a large pot of salted water to a boil. Add the pasta and cook until al dente according to package directions. Reserve 1 cup of pasta water before draining. Drain the pasta and set aside.

While the pasta cooks, heat 2 tablespoons of olive oil in a large skillet over medium-high heat. Add the broccoli florets and diced stem, and cook for 4-5 minutes, stirring occasionally, until bright green and slightly charred. Add the drained chickpeas and cook for another 2-3 minutes until the chickpeas are lightly golden and crispy. Season with a pinch of salt and pepper. Transfer to a plate and set aside.

Reduce heat to medium-low and add the remaining olive oil to the same skillet. Add the sliced garlic and red pepper flakes (if using). Cook for 1-2 minutes, stirring constantly, until the garlic is fragrant and just golden. Be careful not to burn it.

Add the drained pasta to the skillet with the garlic oil. Toss well to coat. Add the reserved broccoli and chickpeas back to the skillet, along with the grated Parmesan and half of the parsley. Drizzle in about 1/2 cup of reserved pasta water and toss vigorously until the sauce is creamy and coats the pasta. Add more pasta water a tablespoon at a time if needed.

Taste and adjust seasoning with salt and pepper. Serve immediately in warm bowls, topped with extra Parmesan, a drizzle of olive oil, and the remaining parsley.