Melt 1 tablespoon of the butter in a large skillet over medium-high heat. Add the mushroom and cook, stirring occasionally, until browned, about 5 minutes. Add the prosciutto and onion and cook, stirring often, until the onion softens, about 2 minutes more. Sprinkle with flour and stir well. Stir in the marsala and broth and bring to a boil. Cook, stirring often, until reduced to about 1 cup, about 5 minutes. Remove from heat to cool slightly, then pour over desired dishes or into a bowl.
Sauce can be kept at room temperature for up to two hours.