Cook the pasta and broccoli.
Bring a large pot of lightly salted water to a boil.
Cook the pasta according to package directions for al dente. During the final 2 minutes of cooking, add the frozen broccoli directly to the pot.
Drain the pasta and broccoli well, return the mixture to the hot pot, and cover to keep warm.
Start the cheese sauce.
While the pasta cooks, whisk together the evaporated milk, flour, and 1/8 teaspoon (40 ml / 1.4 fl oz) of pepper in a medium saucepan.
Cook over medium heat, stirring constantly, until the sauce thickens and begins to bubble gently.
Melt the cheese and combine.
Remove the saucepan from the heat and stir in the shredded cheddar and American cheeses.
Continue stirring until the cheese is completely melted and the sauce is smooth and glossy.
Pour the cheese sauce over the warm pasta and broccoli mixture, stirring gently to coat everything evenly.
Serve immediately, sprinkled with extra black pepper if desired.
Don't overcook the pasta.
Cooking it just to al dente ensures it will hold its shape and not become mushy after being combined with the hot cheese sauce.