Preheat the oven to 375°F (190°C).
Steam the broccoli florets for 4-5 minutes until tender but still crisp. Drain and set aside.
In a medium skillet, melt the butter over medium heat. Add the chopped onion and minced garlic, sautéing for 3-4 minutes until translucent and fragrant.
In a large mixing bowl, combine the heavy cream, softened cream cheese, Dijon mustard, salt, black pepper, and paprika. Whisk until smooth.
Add the sautéed onions and garlic to the cream mixture and stir to combine.
Fold in the shredded chicken, steamed broccoli, cheddar cheese, and half of the mozzarella cheese into the creamy mixture.
Transfer the mixture to a greased 9×13 inch casserole dish, spreading it evenly.
Sprinkle the remaining mozzarella cheese and grated Parmesan cheese evenly over the top.
Bake 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 it rest for 5 minutes before serving.