Preheat the oven to 375°F (190°C).
Steam the broccoli florets until just tender, about 5 minutes, then drain and set aside.
In a medium saucepan over medium heat, melt the butter.
Add the minced garlic to the melted butter and sauté for 1 minute until fragrant.
Add the heavy cream and softened cream cheese to the saucepan, stirring continuously until the cream cheese is fully melted and the mixture is smooth.
Stir in the shredded cheddar cheese, mozzarella cheese, Parmesan cheese, salt, black pepper, and smoked paprika. Continue stirring until the cheese sauce is smooth and creamy.
In a large mixing bowl, combine the shredded cooked chicken, steamed broccoli, and cheese sauce. Mix well to evenly coat the chicken and broccoli.
Transfer the mixture to a greased 9×13-inch baking dish and spread it out evenly.
Bake in the preheated oven for 20-25 minutes, or until the casserole is bubbly and the top is golden brown.
Remove from oven and sprinkle chopped green onions over the top before serving.