Cook pasta until al dente. Add frozen broccoli to the pasta in the last two minutes of cooking.
While the pasta is cooking whisk the evaporated milk, egg yolk, salt, and garlic together in a medium mixing bowl.
Drain pasta and broccoli and return to pan. With the heat on low, add evaporated milk mixture and butter to the pasta. Stir until the pasta is completely coated and then add cheeses, stirring until everything is melted and creamy. If necessary add a little bit of pasta water to loosen the sauce up. Season with salt and pepper.