Preheat the oven to 375°F (190°C).
Bring a large pot of salted water to a boil. Add the broccoli and cauliflower florets and blanch for 3 minutes until slightly tender. Drain and set aside.
In a large skillet, heat the olive oil over medium heat. Add the minced garlic and sauté for 1 minute until fragrant.
Add the softened cream cheese and heavy cream to the skillet. Stir continuously until the cream cheese melts and the mixture is smooth.
Remove the skillet from heat and stir in the shredded cheddar cheese, mozzarella cheese, grated Parmesan cheese, salt, black pepper, and smoked paprika. Mix until well combined.
In a large mixing bowl, combine the shredded cooked chicken, blanched broccoli, and cauliflower.
Pour the cheese sauce over the chicken and vegetables and gently fold to coat everything evenly.
Transfer the mixture to a greased 9×13 inch casserole dish and spread it out evenly.
Bake in the preheated oven for 25 minutes until the casserole is bubbly and the top is golden brown.
Remove from oven and sprinkle chopped green onions over the top before serving.