Preheat the oven to 375°F (190°C).
In a medium saucepan, combine the rinsed jasmine rice and chicken broth. Bring to a boil over medium-high heat, then reduce heat to low, cover, and simmer for 15 minutes or until the rice is tender and liquid is absorbed. Remove from heat and fluff with a fork.
While the rice cooks, 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 to the skillet and cook for 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. Remove from heat.
Steam the broccoli florets until tender but still crisp, about 4-5 minutes. Drain well.
In a large mixing bowl, combine the cooked jasmine rice, cooked chicken and onions, steamed broccoli, sour cream, mayonnaise, shredded cheddar cheese, and shredded mozzarella cheese. Mix well until all ingredients are evenly incorporated.
Transfer the mixture into a greased 9×13-inch baking dish and spread evenly.
Sprinkle the grated Parmesan cheese evenly over the top of the casserole.
Bake uncovered in the preheated oven for 25-30 minutes, or until the casserole is bubbly and the top is golden brown.
Remove from the oven and let it rest for 5 minutes before serving.