Prepare the linguine according to the package directions in a 3-quart saucepan. Add the broccoli during the last 4 minutes of the cooking time. Drain the linguine and broccoli well in a colander.
Heat the butter in a 10-inch skillet over medium-high heat. Add the chicken and cook until it's well browned, stirring often.
Stir the soup, milk, cheese, black pepper and linguine mixture into the skillet. Heat, stirring occasionally, until the mixture is hot and bubbling. Serve with additional Parmesan cheese.