Bring a large pot of salted water to a boil and cook the pasta until just shy of al dente, about 2 minutes less than the package suggests. Reserve 1/2 cup of the pasta cooking water, then drain.
In a large skillet over medium heat, melt the butter and sauté the sliced mushrooms until they release their moisture and begin to brown, about 5 minutes. Season with a pinch of salt and pepper.
Sprinkle the flour over the mushrooms and cook for 1 minute, whisking to avoid lumps. Gradually whisk in the chicken broth and milk, cooking until the mixture thickens into a creamy sauce.
Stir in the heavy cream, Parmesan cheese, thyme, and pepper. Cook until the cheese is melted and the sauce is smooth. Remove from heat.
In a separate bowl, whisk the egg yolk with a splash of the hot sauce to temper it, then whisk the tempered yolk into the cream sauce to brighten the richness.
Fold the shredded chicken, the cooked pasta, and half of the mozzarella into the sauce until evenly coated. If the mixture seems a touch thick, add reserved pasta water a little at a time to reach a creamy consistency.
Preheat your oven to 375°F (190°C). Lightly grease a 2-quart baking dish. Transfer the pasta mixture to the dish and top with the remaining mozzarella and an extra sprinkle of Parmesan.
Bake for 20–25 minutes until the top is bubbling and golden. If you like a deeper crust, broil for an additional 1–2 minutes watching closely to prevent scorching.
Let the dish rest for 5–10 minutes before serving. This helps the sauce set a little so every forkful is creamy without sliding off the pasta.