Preheat the oven to 375°F (190°C). Lightly grease a 9×13-inch casserole dish.
Rinse the jasmine rice under cold water until the water runs clear. In a medium saucepan, combine the 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 let stand covered for 5 minutes.
While the rice cooks, heat olive oil in a large skillet over medium heat. Add the chopped onion and sauté for 3 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, pepper, and paprika, and 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 just tender, about 4-5 minutes. Drain well.
In a large mixing bowl, combine the cooked jasmine rice, cooked chicken and onion mixture, steamed broccoli, sour cream, mayonnaise, shredded cheddar cheese, and half of the mozzarella cheese. Stir until well combined.
Transfer the mixture to the prepared casserole dish and spread evenly.
Sprinkle the grated Parmesan cheese and the remaining mozzarella cheese evenly over the top.
Bake in the preheated oven for 20-25 minutes, or until the casserole is bubbly and the cheese on top is melted and golden brown.
Remove from the oven and let the casserole rest for 5 minutes before serving.