Easy recipe Prosciutto-Stuffed Chicken with Mushroom Sauce

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four skinless, boneless chicken breasts (about about two pounds)Kosher salt, freshly ground pepperabout eight thin slices prosciuttoabout eight thin slices provolone cheesethirty-two fresh basil leavesthree tablespoons unsalted butter, dividedone tablespoon vegetable oilfour cups mixed mushrooms (Recipes by ingridients mushrooms) (such as crimini, chanterelle, and maitake), torn or chopped into small piecesabout two cups low-sodium chicken brothone teaspoon red wine vinegarabout two tablespoons mixed chopped fresh herbs (such as basil, parsley, and chives). .