Prosciutto-Stuffed Chicken with Mushroom Sauce

INGREDIENTS

• Kosher salt, freshly ground pepper
• 2 cups low-sodium chicken broth
• 8 thin slices provolone cheese
• 4 cups mixed mushrooms (such as crimini, chanterelle, and maitake), torn or chopped into small pieces
• 1 teaspoon red wine vinegar
• 32 fresh basil leaves
• 3 tablespoons unsalted butter, divided
• 2 tablespoons mixed chopped fresh herbs (such as basil, parsley, and chives)
• 4 skinless, boneless chicken breasts (about 2 pounds)
• 8 thin slices prosciutto
• 1 tablespoon vegetable oil