Preheat the oven to 375°F (190°C). Lightly grease a 9×13-inch baking dish.
Heat olive oil in a large skillet over medium heat. Add the chopped onion and sauté for 3-4 minutes until translucent.
Add the minced garlic and cook for another 1 minute until fragrant.
Add the chicken pieces to the skillet. Season with salt, black pepper, paprika, and dried thyme. Cook for 6-8 minutes, stirring occasionally, until the chicken is cooked through and no longer pink inside.
Meanwhile, steam the broccoli florets until tender but still crisp, about 4-5 minutes. Drain and set aside.
In a large mixing bowl, combine the Greek yogurt, chicken broth, shredded cheddar cheese, shredded mozzarella cheese, and crushed red pepper flakes (if using). Stir until smooth.
Add the cooked chicken mixture and steamed broccoli to the bowl with the sauce. Mix well to combine all ingredients evenly.
Transfer the mixture into the prepared baking dish and spread evenly.
Sprinkle the grated Parmesan cheese evenly over the top of the casserole.
Bake in the preheated oven for 20-25 minutes, or until the casserole is bubbly and the top is golden brown.
Remove from the oven and let it cool for 5 minutes before serving.