Preheat the oven to 375°F (190°C).
Steam the broccoli florets until just tender, about 4-5 minutes, then drain and set aside.
In a large mixing bowl, combine the sour cream, mayonnaise, ranch seasoning mix, minced garlic, salt, and black pepper. Stir until smooth.
Add the shredded chicken, steamed broccoli, chopped onion, cheddar cheese, and mozzarella cheese to the bowl. Mix well to evenly coat all ingredients with the ranch mixture.
Transfer the mixture to a greased 9×13-inch casserole dish and spread evenly.
In a small bowl, combine the breadcrumbs, grated Parmesan cheese, and melted butter. Mix until the breadcrumbs are evenly coated.
Sprinkle the breadcrumb mixture evenly over the top of the casserole.
Bake in the preheated oven for 30-35 minutes, or until the casserole is bubbly and the breadcrumb topping is golden brown.
Remove from the oven and let it rest for 5 minutes before serving.