Cook the Pasta and Broccoli : Bring a large pot of salted water to a boil. Cook the fettuccine according to package instructions.
During the last 3 minutes of cooking, add the broccoli to the pot.
Before draining, reserve 1 cup of the pasta water. Then, drain and set aside.
Cook the Chicken - While the pasta is cooking, heat 2 tbsp olive oil in a large skillet over medium heat.
Season the chicken with salt and pepper, then add to the skillet. Cook, stirring occasionally, until browned and fully cooked (about 6-8 minutes).
Add the minced garlic and cook for another 30 seconds, then transfer the chicken to a bowl.
Make the Alfredo Sauce - In the same skillet, melt 2 tbsp butter over medium heat.
Stir in 1.5 tbsp flour to form a paste, cooking for about 1 minute.
Slowly whisk in 1 cup milk, stirring constantly until smooth and thickened.
Stir in 1/2 cup heavy cream, all but 1 tbsp of the Parmesan, and a pinch of salt and pepper.
Return the cooked chicken to the pan and stir to combine.
Combine Everything - Add the drained pasta and broccoli to the sauce. Toss everything together, using tongs to coat evenly.
If the sauce is too thick, add a splash of the reserved pasta water until it reaches your desired consistency.
Serve & Enjoy
Divide the Alfredo pasta among four bowls.
Top with the reserved Parmesan cheese, chopped chives, and freshly ground black pepper.
Serve immediately and enjoy!