Bring a large pot of salted water to a boil. Cook the pasta according to package directions until al dente. Reserve 1/2 cup of pasta water, then drain the pasta and set aside.

In a shallow bowl, whisk together the flour, salt, and pepper. Dredge the chicken pieces in the flour mixture, shaking off any excess.

Heat 2 tablespoons of olive oil and 1 tablespoon of butter in a large skillet over medium-high heat. Add the chicken in a single layer and cook until golden brown and cooked through, about 5-7 minutes. Transfer chicken to a plate.

In the same skillet, add the remaining 1 tablespoon of olive oil and 1 tablespoon of butter. Add the sliced mushrooms and sauté undisturbed for 3-4 minutes to brown, then stir and cook for another 4 minutes until liquid has evaporated.

Stir in the minced shallots and garlic. Cook for 1-2 minutes until fragrant and translucent.

Pour in the Marsala wine and chicken broth. Use a wooden spoon to scrape the browned bits (fond) from the bottom of the skillet. Simmer the liquid for 4-5 minutes until it has reduced by approximately half.

Reduce the heat to medium-low. Stir in the heavy cream and the remaining 2 tablespoons of butter. Simmer gently for 2-3 minutes until the sauce begins to thicken.

Whisk in the Parmesan cheese until melted and smooth. Return the cooked chicken and any accumulated juices to the skillet.

Add the cooked pasta to the skillet and toss to coat thoroughly in the sauce. If the sauce is too thick, add the reserved pasta water 1 tablespoon at a time until the desired consistency is reached.

Garnish with chopped fresh parsley and serve immediately with additional Parmesan if desired.