Season each chicken breast with salt and pepper. Place the flour in a flat bowl and dredge the chicken in it, shaking off the excess flour.Heat the oil in a heavy skillet over medium-high heat. Fry the chicken cutlets on both sides until lightly browned, about 4-5 minutes. Place chicken breasts on a plate and tent with aluminum foil to keep warm.
In the same pan, melt the butter over medium heat. Add the mushrooms and cook, stirring occasionally, until the juice released from the mushrooms evaporates and the mushrooms just begin to brown, 4-6 minutes. Add the shallot, garlic, and thyme and cook for 2 minutes. Add the porcini mushrooms and stir to combine. Add the marsala and broth and bring to a boil, scraping any bits from the bottom of the pan. Reduce the heat and continue to simmer for 2 minutes until the liquid is slightly reduced. Add the chicken to the pan and gently coat the chicken with the sauce. Reduce the heat to low and simmer for 3-4 minutes, or until the chicken is heated through and the sauce is slightly thickened. Add salt and pepper to taste. Sprinkle with parsley.
Serve immediately with your choice of sides (e.g., hot pasta). See blog post for other serving suggestions.