Preheat the oven to 350°F (175°C).
Cook the egg noodles according to package instructions until al dente. Drain and set aside.
Steam the broccoli florets until tender but still bright green, about 4-5 minutes. Drain and set aside.
In a large skillet, melt the butter over medium heat. Add the chopped onion and minced garlic, sautéing until the onion is translucent, about 3-4 minutes.
In a large mixing bowl, combine the cooked chicken, steamed broccoli, cooked noodles, sautéed onion and garlic, cream of mushroom soup, sour cream, mayonnaise, milk, salt, black pepper, and paprika. Mix well until all ingredients are evenly incorporated.
Stir in 1/2 cup of shredded cheddar cheese and all of the Parmesan cheese.
Transfer the mixture into a greased 9×13 inch baking dish and spread evenly.
Sprinkle the remaining 1/2 cup shredded cheddar cheese evenly over the top.
Bake uncovered in the preheated oven for 25-30 minutes, or until the casserole is bubbly and the cheese on top is melted and lightly golden.
Remove from oven and let it rest for 5 minutes before serving.