Preheat the oven to 375°F (190°C).
Steam the broccoli florets until just tender, about 5 minutes. Drain and set aside.
In a medium saucepan, melt the butter over medium heat. Add the minced garlic and cook for 1 minute until fragrant.
Add the cream cheese and heavy cream to the saucepan. Stir continuously until the cream cheese is fully melted and the mixture is smooth.
Remove the saucepan from heat and stir in the shredded cheddar cheese, mozzarella cheese, grated Parmesan cheese, salt, black pepper, and paprika until the cheese is melted and the sauce is 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 9×13 inch baking dish and spread it out evenly.
Bake in the preheated oven for 25 minutes or until the casserole is bubbly and the top is golden brown.
Remove from the oven and sprinkle chopped green onions over the top before serving.