Bring a pot of water to a boil and cook the fettuccine according to package instructions, reducing cooking time by 2 minutes. In the last 3 minutes, add broccoli. Drain, reserving some pasta water.
In the same skillet used for chicken, add heavy cream, butter, minced garlic, and parmesan. Stir over medium-low heat until the sauce is creamy.
Add the cooked fettuccine, broccoli, and chicken back into the sauce. Toss until well coated, then remove from heat.