Slice chicken breasts in half widthwise using a sharp knife. Cover with plastic wrap; pound with the flat side of a meat mallet to 1/4-inch thickness. Season both sides with salt and pepper.
Heat 1 tablespoon olive oil and 2 teaspoons butter in a large skillet over medium heat. Add 3 chicken pieces; cook until golden, about 5 minutes per side. Transfer to a large platter. Cook remaining chicken pieces in batches, adding 1 tablespoon olive oil and 2 teaspoons butter for each batch.
Cook and stir shallot and prosciutto in the hot skillet until shallots soften, about 1 minute. Add mushrooms; cook until browned, about 5 minutes. Season with salt and pepper. Pour in Marsala wine; cook until slightly reduced. Pour in chicken stock; simmer until slightly thickened, about 1 minute.
Stir 1 tablespoon butter into the skillet. Return chicken to the skillet; simmer until heated through, about 1 minute. Garnish with chopped parsley.
If making ahead, prepare the chicken through step 2 and refrigerate in a baking dish. When ready to serve, cook remaining ingredients, pour over chicken in the baking dish and bake at 300 degrees F (150 degrees C) for 15 minutes.