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

During the final 2–3 minutes of the pasta cooking time, add the broccoli florets to the boiling water. Cook until bright and crisp-tender, then drain the pasta and broccoli together. When using frozen broccoli, allow a little additional cooking time as needed.

Heat the olive oil in a large skillet over medium heat. Add the garlic and red pepper flakes and cook for 30–60 seconds, stirring constantly, until fragrant. Add the chickpeas, season lightly with salt and black pepper, and cook for 3–4 minutes until hot and lightly golden around the edges.

Pour in the vegetable broth and add the drained broccoli. Bring the liquid to a gentle simmer and cook for 2–3 minutes, stirring occasionally, until the broccoli is tender and the broth has reduced slightly.

Add the drained pasta, lemon juice, and 2 tablespoons reserved pasta water. Toss over medium-low heat for 1–2 minutes until the sauce coats the noodles. Add additional pasta water one tablespoon at a time when needed.

Remove the skillet from the heat. Stir in the Parmesan when using it, then taste and adjust the salt, black pepper, lemon, and red pepper flakes. Serve immediately with optional lemon wedges and a light drizzle of olive oil.