Preheat the oven to 350°F (175°C).
Cook the egg noodles according to package instructions until al dente. Drain and set aside.
In a large skillet, melt the butter over medium heat. Add the diced onion and cook until softened, about 3-4 minutes.
Add the minced garlic and sliced mushrooms to the skillet. Cook for another 5 minutes until mushrooms are tender.
In a large mixing bowl, combine the cooked chicken, cooked noodles, sautéed onion, garlic, and mushrooms, frozen peas, cream of mushroom soup, cream of chicken soup, sour cream, chicken broth, dried thyme, salt, and black pepper. Stir until well combined.
Transfer the mixture to a 9×13-inch baking dish and spread evenly.
Sprinkle the shredded cheddar cheese and grated Parmesan cheese evenly over the top.
Bake uncovered in the preheated oven for 35-40 minutes, or until the casserole is bubbly and the cheese is melted and golden brown.
Remove from the oven and let it rest for 5 minutes before serving.