01 - Bring a large pot of salted water to a boil. Add the pasta and cook according to package instructions, adding the broccoli florets to the pot for the last 3 minutes of cooking. Drain and set aside.
02 - In a large skillet over medium heat, melt the butter. Add the minced garlic and sauté for 1 minute until fragrant.
03 - Sprinkle the flour over the butter and garlic, whisking constantly for 1 minute to make a roux.
04 - Gradually whisk in the milk. Continue to cook, stirring frequently, until the sauce thickens, about 3–4 minutes.
05 - Stir in the Parmesan, salt, black pepper, and red pepper flakes (if using). Cook for another minute until the cheese is melted and the sauce is smooth.
06 - Add the shredded chicken, drained pasta, and broccoli to the skillet. Toss everything together until well coated and heated through, about 2–3 minutes.
07 - Taste and adjust seasoning if needed. Serve hot, garnished with fresh parsley and extra Parmesan.