Bring a large pot of salted water to a boil. Add the 12 oz dry fettuccine and cook according to package directions. Add the 4 cups broccoli florets to the pot during the last 5 minutes of the pasta cooking time so the pasta and broccoli finish cooking together. When done, drain the pasta and broccoli, reserving 1/4 cup of the pasta cooking water; set the drained pasta and broccoli aside.
In a large, deep skillet or pot, melt 2 Tbsp butter over medium heat. Add the 2 minced garlic cloves and 3 Tbsp flour; cook, stirring constantly, for 1 minute.
While whisking constantly, slowly pour in the 1 3/4 cups milk. Continue whisking and cook until the mixture begins to thicken and is smooth.
Add the 3 oz diced cream cheese and the 1/2 cup (1.5 oz) finely shredded parmesan to the milk mixture. Whisk until the cheeses melt and the sauce is smooth (it may look lumpy at first but will smooth as it melts).
Stir in the 3 Tbsp heavy cream, 1 Tbsp lemon zest, and 1 Tbsp fresh lemon juice. Season the sauce with salt and freshly ground black pepper to taste.
Add the drained pasta, broccoli, and the 1 lb grilled, sliced chicken breasts to the sauce. Gently toss to combine and coat everything with the sauce. If the sauce is too thick, add up to the reserved 1/4 cup pasta water a little at a time until you reach the desired consistency.
Serve warm, topped with additional parmesan, extra lemon zest, and freshly ground black pepper as desired.