Preheat your oven to 360 degrees F. pan. Cook the spaghetti in a large pot of salted boiling water according to the package directions al dente. Drain and set aside.
Meanwhile, add the oil to a large skillet placed over high heat. Once the oil is hot, cook the mushrooms until they begin to brown and start losing liquid, stirring frequently. Sprinkle with salt. (see note)
Add the garlic and the chicken to the mushrooms and cook for 2 to 3 minutes until the chicken is mostly cooked. (If you want the sauce to be even thicker, sprinkle 2 heaping tablespoons of flour onto the chicken and mushrooms and stir thoroughly before adding in the liquids.)
Pour in the heavy cream and milk and bring to a boil. Add 1 cup of parmesan cheese and cook on medium-high heat until the sauce begins to thicken, around 3-4 minutes.
Season the sauce with salt and pepper to taste and mix in the cooked spaghetti.
Transfer everything to a 9-inch x 13-inch baking dish. Top with the remaining Parmesan cheese and mozzarella.
Bake for 15-20 minutes until cheese is melted and starts to brown. Remove from the oven and serve warm.
Notes
If you want to kick it up a notch, deglaze the mushrooms and chicken with 1/3 cup of white wine. You can even add some dried porcini powder to the sauce.