Preheat and prep: Heat the oven to 375°F (190°C).
Lightly grease a 9x13-inch baking dish with olive oil.
Cook the pasta: Boil in salted water until just shy of al dente, 1–2 minutes less than package directions. Drain and set aside.
Make the roux: In a large skillet or pot, melt butter over medium heat. Stir in flour and cook 1–2 minutes, stirring, until lightly golden and nutty.
Build the sauce: Add garlic; cook 30 seconds.
Whisk in milk and chicken broth gradually, smoothing out any lumps. Simmer 3–5 minutes until thick enough to coat a spoon.
Add cheese and season: Off heat, stir in 3/4 cup Parmesan and 1 cup mozzarella until melted. Season with Italian seasoning, red pepper flakes (if using), salt, and pepper.
Combine: Add cooked pasta and chicken to the sauce.
Toss until everything is evenly coated.
Assemble: Pour into the baking dish. Top with remaining mozzarella and Parmesan for a thick, cheesy layer.
Bake: Bake 18–22 minutes, until bubbly and lightly golden. For extra color, broil 1–2 minutes at the end, watching closely.
Finish: Rest for 5–10 minutes to set.
Garnish with parsley. Serve warm.