Preheat the oven to 375°F (190°C).
Steam the broccoli and cauliflower florets in a steamer basket over boiling water for 5 minutes until tender but still crisp. Drain and set aside.
In a medium saucepan, melt the butter over medium heat. Add the chopped onion and minced garlic and sauté for 3-4 minutes until softened.
Sprinkle the flour over the onion and garlic mixture and stir constantly for 1-2 minutes to form a roux.
Gradually whisk in the chicken broth, stirring continuously until the sauce thickens, about 3-4 minutes.
Remove the saucepan from heat and stir in the sour cream, mayonnaise, salt, black pepper, dried thyme, and paprika until smooth.
In a large mixing bowl, combine the shredded chicken, steamed broccoli, steamed cauliflower, cheddar cheese, mozzarella cheese, and the creamy sauce. Mix well to combine all ingredients evenly.
Transfer the mixture to a greased 9×13 inch baking dish and spread it out evenly.
Sprinkle the grated Parmesan cheese evenly over the top of the casserole.
Bake uncovered in the preheated oven for 30 minutes until the casserole is bubbly and the top is golden brown.
Remove from the oven and let the casserole rest for 5 minutes before serving.