Fill a large stock pot with water, leaving about 2 inches from the top, and bring to a rapid boil.
Add the broccoli florets to the boiling water and cook for 5 minutes until bright green and just tender; remove with a slotted spoon and set aside.
Return the water to a boil if needed, add the cavatappi, and cook according to package directions until al dente; drain and set aside.
While the pasta cooks, heat the olive oil in a skillet over medium heat, then add the minced garlic and cook about 1–2 minutes until fragrant (do not burn).
Add the cooked broccoli and chicken broth to the skillet, then simmer for about 8–10 minutes, stirring occasionally, until the broccoli is fork-tender; season with salt and pepper to taste.
Add the drained cavatappi to the skillet and toss to combine, allowing the pasta to absorb some of the sauce for a minute.
Sprinkle with grated Parmesan cheese, stir to combine, and serve immediately with extra Parmesan if desired.
