Cook fettuccine al dente according to package directions; during last 4 minutes, add broccoli florets. Drain and keep warm in large serving bowl.
While pasta is cooking, prepare chicken. (see note)
Cut chicken into bite-sized cubes. Add salt and pepper to flour; coat chicken pieces with flour mixture.
Heat 12-inch skillet; add 2 Tbsp. butter. Add chicken pieces and cook until no longer pink. Remove from skillet.
Add another 2 Tbsp. butter to skillet. Add onions and garlic; cook 1 minute. Add cream and cook additional 3 minutes until it reduces slightly. Add cheese and stir well until cheese is blended and melted.
Return chicken pieces to cream mixture; stir to coat. Pour chicken and cream mixture over drained pasta. Stir to coat pasta with sauce.
Serve immediately, sprinkling additional Parmesan cheese on top.
NOTE: Instead of using the uncooked chicken breasts, we grill them and then cut into bite-sized chunks. Gives the dish much more flavor.