Heat oven to 375°F. Lightly butter a 9×13-inch baking dish with 1 tsp of the butter and set aside.
In a large pot or Dutch oven over medium heat, melt the remaining butter. Whisk in the condensed soup, sour cream, and chicken broth until smooth and steamy, 3–4 minutes. Season with garlic powder, onion powder, salt, pepper, and a tiny pinch of nutmeg. Stir in the chicken and reduce heat to low to keep warm.
Bring a large pot of salted water to a boil and cook the pasta until just shy of al dente (about 1 minute less than package time). Reserve 1/2 cup pasta water, then drain.
Fold the drained pasta into the warm sauce. If the mixture seems thick, splash in a little reserved pasta water until creamy. Stir in half of the mozzarella and 2 tablespoons of the Parmesan to enrich the sauce.
Transfer the mixture to the prepared dish, smoothing the top. Sprinkle with the remaining mozzarella and the rest of the Parmesan.
Cover tightly with foil and bake for 20 minutes. Uncover and bake 5–7 minutes more until bubbling. For a lightly browned top, broil 1–2 minutes, watching closely.
Let the casserole rest for 8 minutes before serving so the sauce sets and coats the pasta evenly.