Bring a large pot of salted water to a rolling boil. Add the pasta and cook according to the package directions.

While the pasta cooks, cut the broccoli into small, bite-sized florets.

When the pasta has 2 minutes left, add the broccoli florets to the boiling water with the pasta.

While the pasta and broccoli cook, combine the olive oil, garlic, lemon zest, lemon juice, Parmesan, dried herbs, and red pepper flakes in a jar with a lid.

Scoop out 1 cup of the starchy cooking water and set it aside. Drain the pasta and broccoli together and return them to the pot off the heat.

Add 0.5 cup of the reserved pasta water to the jar of dressing. Seal the lid and shake vigorously until combined.

Pour the dressing and the shredded cheese over the pasta and broccoli. Stir vigorously until the cheese melts and the sauce coats everything evenly, adding more pasta water a splash at a time if it looks dry. Serve immediately with extra Parmesan.