Bring a large pot of salted water to a boil. Add the macaroni and broccoli and cook for 5 minutes; the pasta will only be about halfway cooked. Reserve 1 cup of the cooking water, then drain the pasta and broccoli and return them to the pot.
Meanwhile, sprinkle the chicken with salt and pepper. Heat 1 tablespoon of the butter in a large skillet over medium-high heat. Add the chicken and cook, stirring occasionally, until browned and cooked through, 5 to 7 minutes; set aside.
Put the pot with the pasta and broccoli over low heat. Stir in the milk, flour, mustard, Worcestershire, 1 teaspoon salt and the remaining 1 tablespoon butter. Bring to a simmer and cook, stirring, until the sauce thickens and the pasta is al dente, 3 to 5 minutes. Turn off the heat, add the Cheddar, Pecorino-Romano, Parmesan, cooked chicken and 1/4 cup of the reserved cooking water and stir until the cheese melts. Gradually add the remaining cooking water as needed until the pasta is creamy and saucy. Serve hot.
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