Season the chicken breasts with salt and pepper.
In a Dutch oven, melt 2 tablespoons of butter and add olive oil. Sear the chicken breasts until fully cooked (about 10 minutes), then remove them from the pot. Cut into bite-size pieces and set aside.
In a large pot of salted boiling water, cook the pasta according to the package instructions until al dente. Drain and set aside.
To the same Dutch oven where you cooked the chicken, add 4 tablespoons of butter and let it melt. Add the sliced mushrooms and cook until they start to brown (about 4 minutes)
Add the chopped onion and minced garlic, and cook until the onions are translucent (2-3 minutes).
Sprinkle the flour over the mushrooms and stir well, cooking for a minute.
Slowly add the chicken stock and half and half, stirring continuously to avoid lumps. Let the sauce thicken slightly.
Add 1 cup of shredded mozzarella and the Parmesan cheese. Stir until the cheese is melted and the sauce is smooth. Remove from heat, and season with salt and pepper to taste.
Return the cooked chicken and pasta to the Dutch oven and mix well to coat everything in the sauce.
Preheat your oven to 375°F (190°C), or 170°C if you have a fan oven. Transfer the mixture to a 9x13-inch baking dish. Top with the remaining 1 cup of shredded mozzarella.
Bake for 30 minutes or until the top is bubbling and golden. Allow to cool for a few minutes before serving.