Bring a large pot of salted water to a boil and cook the fettuccine according to package directions; add the broccoli florets during the last 5 minutes of cooking so both finish together.
Reserve 1/4 cup of the pasta cooking water, then drain the pasta and broccoli together and set aside.
In a large deep skillet over medium heat, melt the butter, add the minced garlic and flour, and cook for 1 minute while stirring constantly to form a roux.
Slowly whisk the milk into the roux, whisking vigorously to prevent lumps, and cook until the mixture begins to thicken.
Add the cubed cream cheese and shredded Parmesan, whisking until the cheeses melt and the sauce becomes smooth.
Stir in the heavy cream, lemon zest, and lemon juice, then season with salt and freshly ground black pepper to taste.
Add the drained pasta, broccoli, and sliced grilled chicken to the sauce and toss to combine; if the sauce is too thick, thin with reserved pasta water a little at a time until desired consistency is reached.
Serve warm topped with additional Parmesan, extra lemon zest, and more black pepper if desired.