Place chicken cutlets between 2 sheets of plastic wrap on a solid, level surface. Firmly pound the meat with the smooth side of a meat mallet to a 1/4-inch thickness. Season both sides with salt and pepper.

Heat 1 tablespoon olive oil and 1 tablespoon butter in a large skillet over medium-high heat until butter is melted. Place chicken cutlets into the skillet; cook until both sides are golden brown, about 5 minutes for each side.

Lower the heat to medium; add olive oil to the skillet if needed. Place prosciutto and shallots into the skillet; cook and stir until warmed, about 1 minute. Stir in the mushrooms and cook until browned, about 5 minutes. Season with salt and pepper. Pour wine into skillet; simmer until flavors combine, about 1 minute. Add chicken stock; simmer and stir until the sauce reduces slightly, about 1 minute.

Stir 1 tablespoon of butter into the skillet. Return chicken to the pan; simmer until chicken is heated through, about 1 minute. Garnish with parsley, salt, and pepper.