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. Taste and adjust seasoning if needed. Serve hot, garnished with fresh parsley and extra Parmesan.