Cook the pasta according to the package directions. Drain.
Meanwhile, bring a pot of salted water to a boil and cook the broccoli until tender, 3 minutes. Drain.
Heat the milk and butter in a large saucepan over low heat and slowly whisk in the flour. Simmer until slightly thickened, whisking constantly, 1 to 2 minutes.
Remove from heat and stir in the Parmesan and ½ teaspoon salt. Add the pasta and broccoli and cook, stirring, over low heat until heated through.
Top each serving with extra Parmesan.