Place the chicken breasts between two sheets of plastic wrap and gently pound them to an even thickness of 1/2-inch.

Spread the flour on a sheet of wax paper, season liberally with salt and pepper and combine well. Dredge the chicken breasts in the flour mixture , shake off the excess and reserve 1 tablespoon of the flour for thickening the sauce.

Heat 1 tablespoon of the butter along with 2 tablespoons of the olive oil in a large skillet over medium-high heat. Add the chicken in a single layer and cook until golden, 2-1/2 to 3 minutes per side. Transfer to a plate and set aside.

Add the remaining 2 tablespoons of butter and 1 tablespoon of olive oil to the pan. Lower the heat to medium and add the shallots and garlic. Sauté until fragrant, 1 minute.

Add the mushrooms, season with salt and pepper and continue cooking, stirring frequently, until any moisture they give off has evaporated. Sprinkle the remaining tablespoon of flour over the mixture, combine well and continue cooking until the mushrooms are lightly browned, 3 to 4 minutes.

Add the Marsala and chicken broth to the pan and continue cooking, stirring frequently, until the mixture begins to thicken. Taste and adjust the seasoning as needed, then return the chicken to the pan for 1 to 2 minutes, turning the pieces several times to coat with the sauce.

Transfer the chicken to a platter and spoon the sauce over top. Sprinkle with chopped parsley and serve immediately.