Start by bringing a large pot of salted water to a boil. Add the fettuccine pasta and cook according to package instructions until al dente. Once cooked, reserve about 1 cup of the pasta water, then drain the pasta and set it aside.
In the same pot, add the broccoli florets to the boiling water for about 2-3 minutes, just until tender and bright green. Drain the broccoli and set aside with the pasta.
While the pasta and broccoli are cooking, heat your grill or grill pan over medium-high heat. Season the chicken breasts with salt and pepper, then grill for about 6-7 minutes on each side, or until fully cooked. Remove from the grill and slice into strips.
In a large skillet, melt the butter over medium heat. Add the minced garlic and sauté for about 1 minute until fragrant. Stir in the flour and cook for another minute, forming a roux.
Gradually whisk in the milk, ensuring there are no lumps. Cook for about 3-4 minutes until the sauce begins to thicken. Add the cream cheese, stirring until melted and combined.
Stir in the shredded Parmesan cheese, heavy cream, lemon zest, and lemon juice. Season with salt and freshly ground black pepper to taste. If the sauce is too thick, add reserved pasta water a little at a time until you reach your desired consistency.
Add the cooked fettuccine and broccoli to the Alfredo sauce, tossing to coat evenly. Finally, add the grilled chicken strips on top and gently mix everything together.
Divide the Lemon Fettuccine Alfredo with Grilled Chicken and Broccoli among plates. Top with additional Parmesan cheese and lemon zest for a fresh finish.