Preheat the oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat. Add diced onion and cook for 4-5 minutes until softened.
Add minced garlic to the skillet and cook for an additional 1 minute until fragrant.
Add sliced zucchini to the skillet, season with salt, pepper, and dried Italian seasoning. Cook for 5-7 minutes until zucchini is just tender. Remove from heat.
In a large mixing bowl, combine shredded rotisserie chicken, cooked zucchini mixture, sour cream, mayonnaise, mozzarella cheese, cheddar cheese, and half of the Parmesan cheese. Mix well.
Transfer the mixture into a greased 9×13-inch baking dish and spread evenly.
Sprinkle the remaining Parmesan cheese and breadcrumbs evenly over the top of the casserole.
Bake in the preheated oven for 30-35 minutes, or until the casserole is bubbly and the topping is golden brown.
Remove from the oven and let it cool for 5 minutes before serving.