Preheat the oven to 375°F (190°C). Lightly grease a 9×13-inch baking dish.
Cook the ziti pasta according to package instructions until al dente. Drain and set aside.
Steam the broccoli florets until just tender, about 4-5 minutes. Drain and set aside.
Heat olive oil in a large skillet over medium-high heat. Add the diced chicken, garlic powder, onion powder, salt, and black pepper. Cook until the chicken is cooked through and lightly browned, about 6-8 minutes. Remove from heat.
In a medium saucepan, melt the butter over medium heat. Whisk in the flour and cook for 1-2 minutes until lightly golden and bubbling.
Gradually whisk in the milk, continuing to whisk until the sauce thickens and comes to a simmer, about 5-7 minutes.
Remove the sauce from heat and stir in salt, black pepper, ground nutmeg, 1 cup mozzarella cheese, 1/2 cup Parmesan cheese, and shredded cheddar cheese until melted and smooth.
In a large mixing bowl, combine the cooked ziti, steamed broccoli, cooked chicken, diced onion, minced garlic, and the cheese sauce. Stir until everything is evenly coated.
Transfer the mixture to the prepared baking dish. Sprinkle the remaining 1/2 cup mozzarella cheese and 1/2 cup Parmesan cheese evenly over the top.
Bake uncovered in the preheated oven for 25-30 minutes, or until the casserole is bubbly and the cheese on top is melted and golden brown.
Remove from the oven and let rest for 5 minutes before serving.