Pound the chicken to an even thickness, about 1cm thick, for even cooking.
Season the chicken with salt and pepper, then lightly dredge in flour, shaking off excess.
Heat a large non-stick pan over medium-high heat. Add half the oil and butter. Cook the chicken for 3-4 minutes per side until golden brown. Remove and set aside.
In the same pan, add the remaining butter and oil. Sauté shallots and garlic for about 1 minute until fragrant.
Add mushrooms and cook for about 3 minutes until lightly browned.
Pour in Marsala wine and bring to a boil. Reduce by half over high heat, about 3 minutes.
Lower heat and add chicken broth, heavy cream, salt, and pepper. Simmer for 3-5 minutes until thickened.
Return chicken to the pan, letting it simmer in the sauce for about 1 minute.
Garnish with parsley and serve with pasta, mashed potatoes, or crusty bread.