Preheat the oven to 350°F (175°C).
Bring a large pot of salted water to a boil. Add the broccoli florets and blanch for 2 minutes until bright green and slightly tender. Drain and set aside.
In a large skillet, heat 2 tablespoons olive oil over medium heat. Add the diced onion and cook for 3-4 minutes until translucent.
Add the minced garlic to the skillet and cook for 1 minute until fragrant.
Add the ground beef to the skillet. Cook, breaking it apart with a spoon, until browned and cooked through, about 6-8 minutes. Drain any excess fat.
Add the diced chicken pieces to the skillet with the beef and cook for 5-7 minutes until the chicken is no longer pink inside.
Season the meat mixture with salt, black pepper, dried thyme, and paprika. Stir well to combine.
In a large mixing bowl, combine the cream of mushroom soup, sour cream, mayonnaise, and chicken broth. Mix until smooth.
Add the cooked meat mixture and blanched broccoli to the bowl with the sauce. Stir gently to combine everything evenly.
Transfer the mixture to a greased 9×13-inch baking dish. Spread evenly.
Sprinkle the shredded cheddar cheese, mozzarella cheese, and grated Parmesan cheese evenly over the top.
Bake uncovered in the preheated oven for 30 minutes, or until the casserole is bubbly and the cheese is melted and golden brown.
Remove from the oven and let cool for 5 minutes before serving.