Preheat the oven to 375°F (190°C).
Bring a large pot of salted water to a boil. Add the broccoli florets and cook for 2 minutes until bright green and slightly tender. Drain and set aside.
Heat olive oil in a large skillet over medium-high heat. Add the chopped onion and garlic; sauté for 3 minutes until softened.
Add the ground beef to the skillet. Cook, breaking it up with a spoon, until browned and cooked through, about 6-8 minutes. Remove the beef from the skillet and set aside.
In the same skillet, add the diced chicken pieces. Cook for 6-7 minutes until the chicken is cooked through and lightly browned. Remove and set aside with the beef.
Reduce heat to medium and add the flour to the skillet. Stir constantly for 1-2 minutes to cook the flour without browning.
Gradually whisk in the milk and chicken broth, stirring continuously until the sauce thickens, about 4-5 minutes.
Add salt, black pepper, dried thyme, and paprika to the sauce. Stir well.
Remove the skillet from heat and stir in 1 1/2 cups of shredded cheddar cheese until melted and smooth.
In a large mixing bowl, combine the cooked ground beef, cooked chicken, blanched broccoli, cooked rice, and cheese sauce. Mix gently until well combined.
Transfer the mixture to a greased 9×13-inch baking dish.
Sprinkle the top evenly with the remaining 1/2 cup cheddar cheese, shredded mozzarella cheese, and grated Parmesan cheese.
Bake uncovered in the preheated oven for 25-30 minutes, or until the casserole is bubbly and the cheese on top is golden brown.
Remove from the oven and let rest for 5 minutes before serving.