Cook pasta according to package directions; reserve ½ cup pasta cooking water before draining; set aside.
Steam the broccoli for 2 minutes until fork tender, and toss with the cooked pasta.
Heat 1 tablespoon butter in a large skillet over medium high heat. Season chicken breasts with ½ teaspoon salt and ¼ teaspoon black pepper. Add to skillet and cook until cooked through, about 3-4 minutes per side. Toss with the cooked pasta and broccoli.
In the same pan, heat the remaining butter, add garlic and cook, stirring constantly, until fragrant, about 1 minute.
Add flour, then whisk to combine and cook for 1 additional minute. Slowly whisk in the milk and season with the remaining salt and pepper. Let the sauce simmer uncovered until thickened, about 2-3 minutes. Turn off the heat, and fold in the parmesan cheese.
Pour the sauce over the pasta, broccoli and chicken, and stir well. Add as much of the reserved pasta water for the desired sauce thickness. Serve with more parmesan cheese, if desired.
Video
Notes
Storage: Store leftovers in an airtight container with a lid and keep them in the fridge. This should keep them good for 2-3 days.