Preheat the oven to 375°F (190°C). Lightly grease a 9×13 inch baking dish.
Bring a large pot of salted water to a boil. Add the ziti pasta and cook according to package instructions until al dente, about 8-9 minutes. During the last 3 minutes of cooking, add the chopped broccoli florets to the boiling pasta water. Drain pasta and broccoli and set aside.
While pasta cooks, heat olive oil in a large skillet over medium-high heat. Add the chicken pieces, season with salt, black pepper, and Italian seasoning. Cook until chicken is cooked through and lightly browned, about 6-8 minutes. Add minced garlic and cook for an additional 1 minute until fragrant. Remove from heat.
In a medium saucepan, melt 3 tablespoons of butter over medium heat. Whisk in the flour and cook for 1-2 minutes, stirring constantly, to form a roux.
Gradually whisk in the milk and chicken broth, continuing to whisk until the sauce thickens and comes to a gentle boil, about 5-7 minutes.
Remove the sauce from heat and stir in 1 cup shredded mozzarella cheese, 1/4 cup Parmesan cheese, and shredded cheddar cheese until melted and smooth. Season with salt and crushed red pepper flakes if using.
In a large mixing bowl, combine the cooked pasta and broccoli, cooked chicken, and cheese sauce. Mix well to evenly coat all ingredients.
Transfer the mixture to the prepared baking dish. Sprinkle the remaining 1/2 cup mozzarella cheese and 1/4 cup Parmesan cheese evenly over the top.
Bake uncovered in the preheated oven for 20-25 minutes, until the casserole is bubbly and the cheese on top is melted and golden brown.
Let the casserole rest for 5 minutes before serving. Enjoy warm.