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.
Heat olive oil in a large skillet over medium heat. Add diced onion and cook for 3-4 minutes until translucent.
Add minced garlic to the skillet and cook for 1 minute until fragrant.
Add ground beef to the skillet and cook until browned and cooked through, breaking it apart with a spoon. Drain excess fat.
In a large mixing bowl, combine the cooked ground beef mixture, shredded chicken, blanched broccoli, cream of mushroom soup, sour cream, mayonnaise, chicken broth, salt, black pepper, dried thyme, and crushed red pepper flakes. Stir until well combined.
Transfer the mixture to a greased 9×13 inch baking dish.
Sprinkle 1/2 cup of shredded cheddar cheese and the grated Parmesan cheese evenly over the top.
Bake uncovered in the preheated oven for 30 minutes.
Remove the casserole from the oven and sprinkle the remaining 1/2 cup shredded cheddar cheese on top.
Return the casserole to the oven and bake for an additional 10 minutes, or until the cheese is melted and bubbly.
Let the casserole rest for 5 minutes before serving.