Fill a large stock pot with water, leaving about 2 inches from the top, and bring it to a rapid boil.
Add the broccoli florets to the boiling water and cook for 5 minutes until bright green and starting to become tender; remove with a slotted spoon and set aside.
Return the pot to a boil if needed, add the cavatappi and cook according to package directions until al dente; drain in a colander and set aside.
While the pasta cooks, heat the olive oil in a large skillet over medium heat. Add the minced garlic and cook, stirring, about 1–2 minutes until fragrant (do not let it brown).
Add the cooked broccoli and the chicken broth to the skillet and simmer for about 10 minutes, stirring occasionally, until the broccoli is fork-tender and the flavors meld; season with salt and pepper to taste.
Add the drained cavatappi to the skillet and toss gently to combine and heat through.
Sprinkle with grated Parmesan cheese, mix well, and serve immediately with extra Parmesan if desired.