Place the chicken breasts in between sheets of plastic wrap and pound thin (¼” thick) with a meat mallet. Add the flour to a shallow bowl or plate and season heavily with salt and pepper.
Heat the oil in a large saute pan over medium-high heat. When the oil is hot, dredge both sides of the chicken breasts in flour, shaking off any excess. Add the chicken to the pan, working in batches if the chicken is too large to fit more than one. Cook for 5 minutes per side, flipping once, until the chicken is golden brown. Move chicken to a sheet pan or platter to keep warm.
Lower the heat to medium and add the prosciutto. Cook for 1 minute to render some fat. Next, add the shallots and mushrooms and cook for 5 minutes or until nicely browned. Season with salt and pepper.
Deglaze with the Marsala and reduce by half to cook off the alcohol. Add the chicken stock and reduce by half again. Add the butter and stir to combine, simmering gently for 1 minute. Season with salt and pepper if needed.
Add the chicken back to the pan and simmer for 2 minutes. Garnish with the chopped parsley and serve.