Cook the Pasta:
Bring a large pot of salted water to a boil. Add the fettuccine pasta and cook according to package instructions until al dente. Reserve 1 cup of pasta water, then drain the pasta and set it aside.

Blanch the Broccoli (Optional):
While the pasta is cooking, bring another pot of water to a boil. Add the broccoli florets and cook for 2-3 minutes, until tender-crisp. Drain and set aside.

Cook the Chicken:
Heat olive oil in a large skillet over medium-high heat. Season the chicken breasts with salt and pepper on both sides. Cook the chicken for 6-7 minutes per side, until golden brown and cooked through. Remove the chicken from the pan, let it rest for a few minutes, then slice it into strips.

Prepare the Alfredo Sauce:
In the same skillet, melt butter over medium heat. Add the minced garlic and sauté for 1 minute until fragrant. Pour in the heavy cream, stirring constantly, and bring it to a simmer. Stir in the grated Parmesan cheese and cook, continuing to stir, until the sauce is smooth and thickened.

Combine Pasta and Sauce:
Add the cooked pasta and broccoli (if using) to the skillet with the Alfredo sauce. Toss to coat evenly. If the sauce is too thick, add some reserved pasta water to adjust the consistency.

Assemble and Serve:
Serve the pasta on plates and top with sliced chicken. Garnish with extra Parmesan and fresh parsley, if desired. Enjoy immediately!