Preheat the oven to 350°F. Cook the rice according to package directions. Set aside.
To blanch the broccoli, bring a large pot of lightly salted water to a boil. Add the broccoli and cook for 2 minutes, until bright green in color. Drain in a colander and rinse thoroughly with cold water. Set aside to dry.
Into a large bowl, add the cooked rice, blanched broccoli, 1/2 cup mozzarella cheese, and 1/2 cup cheddar cheese. Gently stir just until combined.
Into a medium saucepan over medium heat, add the butter. When the butter, add the onion. Cook for two minutes, stirring frequently until the onion begins to soften. Add the garlic and continue cooking for an additional 30 seconds.
Whisk in the flour; the mixture will be thick.
Season with salt, black pepper, paprika, and cayenne pepper, if using.
Gradually add the milk and broth, whisking continuously until the consistency is smooth.
Reduce the heat to low and simmer for 3 minutes until thickened. Remove from heat.
Stir in 1/2 cup of the mozzarella cheese and 1/2 cup of the cheddar cheese. Continue stirring until the cheeses are melted.
Pour the cheese sauce over the rice/broccoli mixture, stirring gently just until fully coated.
Spread the mixture into a greased 8x8-inch casserole dish.
Into a small skillet over medium heat, add the remaining 1 tablespoon of butter. When the butter is melted, add the Panko breadcrumbs. Cook for 2 minutes, stirring frequently just until lightly toasted. Sprinkle the breadcrumbs in an even layer over the casserole mixture.
Bake for 35 minutes, until lightly browned around the edges, then serve hot.