Preheat the oven to 350°F (175°C). Lightly grease a 9×13 inch casserole dish.
Bring a large pot of water to a boil. Add the broccoli florets and cook for 2-3 minutes until bright green and slightly tender. Drain and immediately rinse under cold water to stop cooking. Set aside.
In a large skillet, heat olive oil over medium heat. Add the chopped onion and cook for 3-4 minutes until softened. Add the minced garlic and cook for 1 minute more until fragrant. Remove from heat.
In a large mixing bowl, combine the cream of mushroom soup, sour cream, mayonnaise, milk, salt, black pepper, and dried thyme. Stir until smooth.
Add the cooked chicken, blanched broccoli, cooked onion and garlic mixture, shredded cheddar cheese, and shredded mozzarella cheese to the bowl. Mix well to combine all ingredients evenly.
Transfer the mixture into the prepared casserole dish and spread it out evenly.
In a small bowl, mix the panko breadcrumbs with the melted butter until the crumbs are evenly coated.
Sprinkle the buttered panko breadcrumbs evenly over the top of the casserole.
Bake in the preheated oven for 35-40 minutes, or until the casserole is bubbly and the panko topping is golden brown.
Remove from the oven and let it rest for 5 minutes before serving.