Bring a large pot of salted water to a boil and cook the pasta according to package directions until al dente. Reserve 1 cup of pasta water, then drain the pasta.

While the pasta cooks, steam or blanch the broccoli florets for 2 to 3 minutes until bright green and just tender. Transfer to ice water briefly to keep the color vibrant, then drain.

In a large skillet over medium heat, melt the butter. When it begins to foam and release a nutty aroma, add the minced garlic and cook for 30 to 60 seconds until fragrant.

Sprinkle the flour over the butter and garlic, stirring constantly to form a roux. Cook for about 1 minute to remove the raw flour taste.

Slowly whisk in the milk and cream, smoothing out any lumps. The sauce will thicken and shimmer; if it becomes too thick, whisk in a splash of reserved pasta water to loosen.

Stir in the Parmesan cheese until melted and silky. Season with lemon zest, lemon juice, red pepper flakes, salt, and pepper. Taste and adjust seasoning; the lemon should lift the richness.

Add the shredded rotisserie chicken and steamed broccoli to the sauce, folding gently to combine. Warm through for 2 to 3 minutes.

Toss the drained pasta into the skillet, adding a little reserved pasta water to reach your desired sauce cling. The sauce should coat each noodle like a soft blanket.

Finish with chopped parsley or basil and an extra grating of Parmesan. Serve immediately, passing extra pepper at the table.