Heat the oven to 375°F, grease a 9-by-13-inch baking dish and boil the ziti 3 minutes short of tender.
Add the broccoli to the pasta pot for the final 2 minutes, then drain both thoroughly.
Melt the butter in a saucepan, cook the garlic for 30 seconds and whisk in the flour for 1 minute.
Whisk in the warm milk, simmer until thickened and stir in the salt, black pepper and 1 cup Parmesan.
Fold the pasta, broccoli and cooked chicken into the sauce and spread the mixture in the dish.
Top with mozzarella and remaining Parmesan and bake until bubbling, 20 to 25 minutes.