Preheat oven to 350°F (175°C).
Cook spaghetti according to package instructions until al dente. Drain and set aside.
In a large skillet, melt butter over medium heat. Add sliced mushrooms and cook until softened and browned, about 5 minutes.
Add chopped onion to the skillet and cook for 3 minutes until translucent.
Stir in minced garlic and cook for 1 minute until fragrant.
Sprinkle flour over the mushroom mixture and stir constantly for 2 minutes to cook the flour.
Gradually whisk in chicken broth and milk, continuing to whisk until the sauce thickens, about 5 minutes.
Remove skillet from heat and stir in Parmesan cheese, salt, black pepper, and dried thyme until cheese is melted and sauce is smooth.
Add cooked chicken, thawed peas, and cooked spaghetti to the sauce. Stir gently to combine everything evenly.
Transfer the mixture to a greased 9×13-inch baking dish. Sprinkle shredded mozzarella cheese evenly on top.
Cover the dish tightly with aluminum foil. At this point, you can refrigerate the casserole for up to 24 hours before baking.
When ready to bake, preheat oven to 350°F (175°C) if not already heated.
Bake covered for 25 minutes. Remove foil and bake uncovered for an additional 10 minutes until cheese is bubbly and golden.
Remove from oven and let rest for 5 minutes. Garnish with chopped fresh parsley before serving.