Bring a large pot of water to a boil. Add the fettuccine and cook for 5–8 minutes, or until al dente. Reserve 1/2 cup of the pasta water, then drain the pasta and set aside.
While the pasta cooks, melt the butter in a large skillet or sauté pan over medium heat. Add the garlic and cook for about 1 minute, stirring, until fragrant.
Whisk in the flour and cook for about 1 minute to form a smooth paste (roux). Slowly pour in the milk while whisking constantly to avoid lumps. Bring to a gentle simmer and cook for 5–6 minutes, stirring frequently, until the sauce thickens slightly.
Add the broccoli florets to the sauce and continue cooking for 4–5 minutes, or until the broccoli is tender. If the sauce becomes too thick, gradually stir in some of the reserved pasta water.
Reduce heat to low. Stir in the Parmesan cheese, salt, pepper, and nutmeg. Mix well until the cheese melts and the sauce is smooth.
Add the drained fettuccine to the pan and toss gently to coat every strand in the creamy sauce. Adjust the consistency with a bit more pasta water if needed.
Taste and season again if necessary. Serve immediately, with extra Parmesan on top if desired.