Preheat the oven to 350°F (175°C).
Place the chicken breasts in a pot and cover with water. Bring to a boil, then reduce heat and simmer for 15 minutes or until cooked through. Remove chicken from water and shred using two forks. Set aside.
In a large skillet, heat olive oil over medium heat. Add diced onion and minced garlic; sauté for 3-4 minutes until softened.
Add ground beef to the skillet. Cook, breaking it up with a spoon, until browned and cooked through, about 6-8 minutes. Drain excess fat.
Add broccoli florets to the skillet and cook for 3-4 minutes until slightly tender.
In a large mixing bowl, combine sour cream, condensed cream of mushroom soup, milk, salt, black pepper, dried thyme, and paprika. Stir until smooth.
Add the cooked ground beef mixture and shredded chicken to the bowl. Mix well to combine all ingredients evenly.
Transfer the mixture to a greased 9×13-inch baking dish. Spread evenly.
Sprinkle shredded cheddar cheese, mozzarella cheese, and grated Parmesan cheese evenly over the top.
Bake uncovered in the preheated oven for 25-30 minutes, or until the casserole is bubbly and the cheese is melted and golden brown.
Remove from oven and let cool for 5 minutes before serving.