Preheat your oven to 375°F (190°C).
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 transfer to a bowl of ice water to stop cooking. Drain again and set aside.
In a large skillet, heat olive oil over medium heat. Add the chopped onion and minced garlic and sauté until softened and fragrant, about 3-4 minutes.
In a large mixing bowl, combine the cooked shredded chicken, blanched broccoli, sautéed onion and garlic, Greek yogurt, chicken broth, Dijon mustard, salt, black pepper, and paprika. Stir well to combine.
Add the shredded cheddar cheese and half of the mozzarella cheese to the mixture and stir until evenly distributed.
Transfer the mixture into a greased 9×13-inch baking dish and spread evenly.
Sprinkle the remaining mozzarella cheese and grated Parmesan cheese evenly over the top.
Bake uncovered in the preheated oven for 25-30 minutes or until the casserole is bubbly and the cheese on top is melted and golden brown.
Remove from the oven and let it cool for 5 minutes before serving.