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 pasta and broccoli 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, stirring constantly, for 1 minute to form a roux.
While whisking, slowly pour the milk into the roux. Cook, whisking vigorously, until the mixture begins to thicken and is smooth.
Add the cubed cream cheese and shredded Parmesan to the sauce and whisk until the cheeses melt and the sauce becomes smooth.
Stir in the heavy cream, lemon zest, and lemon juice. Season with salt and plenty of freshly ground black pepper to taste.
Add the drained fettuccine, broccoli, and sliced grilled chicken to the sauce and toss to combine; if the sauce is too thick, thin with reserved pasta water a tablespoon at a time until desired consistency is reached.
Serve warm, topped with additional Parmesan, extra lemon zest, and more black pepper if desired.
