Preheat the oven to 350°F (175°C). Lightly grease a 9×13-inch baking dish with cooking spray.
Cook the spaghetti according to package instructions until al dente. Drain and set aside.
In a large skillet, melt the butter over medium heat. Add the sliced mushrooms and cook until they release their moisture and begin to brown, about 5 minutes.
Add the chopped onion and cook until softened, about 3 minutes. Stir in the minced garlic and cook for 1 minute until fragrant.
Sprinkle the flour over the mushroom mixture and stir constantly for 2 minutes to form a roux.
Gradually whisk in the chicken broth and milk, stirring continuously to avoid lumps. Cook until the sauce thickens, about 5 minutes.
Remove the skillet from heat and stir in 3/4 cup of the Parmesan cheese until melted and smooth.
Add the shredded chicken, salt, black pepper, thyme, and parsley to the sauce. Stir in the Greek yogurt until combined.
In a large mixing bowl, combine the cooked spaghetti and the chicken sauce mixture. Mix well to coat the pasta evenly.
Transfer the mixture to the prepared baking dish. Sprinkle the remaining 1/4 cup Parmesan cheese and shredded mozzarella evenly over the top.
Cover the dish tightly with aluminum foil and bake for 25 minutes.
Remove the foil and bake for an additional 15 minutes, or until the top is golden and bubbly.
Allow the casserole to cool completely before freezing. To freeze, wrap the dish tightly with plastic wrap and then aluminum foil. Label and freeze for up to 3 months.
To reheat from frozen, thaw overnight in the refrigerator. Bake uncovered at 350°F (175°C) for 30-35 minutes until heated through.