Preheat the oven to 375°F (190°C).
Heat olive oil in a skillet over medium heat. Add diced onion and minced garlic; sauté until the onion is translucent, about 3-4 minutes.
In a large mixing bowl, combine the cooked shredded chicken, steamed broccoli florets, sautéed onion and garlic, Greek yogurt, chicken broth, dried thyme, salt, black pepper, cheddar cheese, and Parmesan cheese. Mix well until all ingredients are evenly incorporated.
Transfer the mixture to a greased 9×13 inch casserole dish and spread it out evenly.
Sprinkle the 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 breadcrumbs are golden brown.
Remove from the oven and let it cool for 5 minutes before serving.