Melt 1 tablespoon of butter in a large skillet over medium-high heat. Add mushrooms 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. Sprinkle with the flour and stir well. Stir in the Marsala and broth, and bring to a boil. Cook for about 5 minutes, stirring often, until reduced to approximately 1 cup. Remove from heat. Add the remaining butter, one tablespoon at a time, and stir until incorporated into the sauce. Season with salt and pepper.