Bring a large pot of salted water to a boil. Cook broccoli until tender and bright green, 2 minutes. Using a slotted spoon, transfer to a colander, leaving water in pot. Add chicken to pot; cook until opaque, 2 to 4 minutes. Transfer to colander with broccoli.
Cook pasta in same pot of water until just tender to the bite, about 10 minutes. Reserve 1/2 cup pasta water; drain pasta.
Simmer cream, reserved 1/2 cup water, and 1/2 cup parmesan in same pot over medium heat, stirring often. Add broccoli, chicken, and pasta, stirring to combine. Season to taste with salt and pepper. Serve with additional parmesan on the side.