Place the chicken breasts in a large ziplock bag and flatten to ½-inch even thickness.
Combine the flour, cornstarch, pepper and salt together in a shallow bowl, pie plate or platter. Set aside.
Heat a large non-stick skillet over medium-high heat. Add 3 tablespoons of the olive oil. Once shimmering, dredge the prepared chicken breasts in the flour mixture shaking off any excess. Sauté the chicken in (two or three) batches until golden brown on each side taking care not to crowd the pan. Remove the cooked chicken to a platter or sheet pan in a single layer. Keep warm.
Lower the heat under the skillet to medium and add the remaining 1 tablespoon of olive oil. Add the prosciutto and sauté for a minute or two to render some of the fat. Add the shallots and garlic and sauté for about 3 minutes or until the garlic is fragrant and the shallots are softening. Add the mushrooms and cook until their moisture releases, then evaporates. Season with fresh ground black pepper and add the thyme.
Add the wine to the pan with the mushrooms and deglaze (scrape up any crunchy bits off the bottom.) Bring to a boil and add the chicken broth. Bring the Marsala sauce to a low simmer and reduce the sauce slightly. Add the butter and stir until melted.
Return the chicken breasts to the skillet and simmer for a few minutes or until the chicken is heated and cooked through. Check seasoning and add salt and pepper if needed. Remove to a platter and garnish with parsley to serve.