Prosciutto-Stuffed Chicken with Mushroom Sauce

Name: Prosciutto-Stuffed Chicken with Mushroom Sauce
Cuisine: European, American, Italian, Mediterranean
Ingredients: 4 skinless, boneless chicken breasts (about 2 pounds); Kosher salt, freshly ground pepper; 8 thin slices prosciutto; 8 thin slices provolone cheese; 32 fresh basil leaves; 3 tablespoons unsalted butter, divided; 1 tablespoon vegetable oil; 4 cups mixed mushrooms (such as crimini, chanterelle, and maitake), torn or chopped into small pieces; 2 cups low-sodium chicken broth; 1 teaspoon red wine vinegar; 2 tablespoons mixed chopped fresh herbs (such as basil, parsley, and chives)