Preheat the oven to 375°F (190°C).
Place the diced potatoes in a large pot and cover with water. Bring to a boil and cook for 8-10 minutes until just tender. Drain and set aside.
Steam the broccoli florets for 4-5 minutes until bright green and slightly tender. Drain and set aside.
In a large skillet, melt the butter over medium heat. Add the chopped onion and minced garlic and sauté for 3-4 minutes until softened and fragrant.
In a large mixing bowl, combine the condensed cream of chicken soup, sour cream, milk, salt, black pepper, dried thyme, and paprika. Stir until smooth.
Add the cooked potatoes, steamed broccoli, cooked chicken, and sautéed onion and garlic to the bowl with the sauce. Mix gently until all ingredients are evenly coated.
Transfer the mixture to a greased 9×13-inch baking dish. Spread evenly.
Sprinkle the shredded cheddar and mozzarella cheese evenly over the top of the casserole.
Bake uncovered in the preheated oven for 30-35 minutes, until the cheese is melted, bubbly, and golden brown.
Remove from the oven and let the casserole rest for 5 minutes before serving.