Halve chicken breasts horizontally, keeping 1 long side attached; open halves like a book. Pound breasts until ¼-inch thick. Season on all sides with salt and pepper.

Place 1 chicken breast on a work surface, cut side up. Overlap 2 slices prosciutto on top of chicken, leaving a ½-inch border. Top prosciutto with 2 slices provolone; layer 8 basil leaves over cheese, maintaining ½-inch border. Roll up chicken lengthwise and tie with kitchen twine. Repeat with remaining chicken breasts.

Preheat oven to 450°. Heat 1 tablespoon butter and oil in a large heavy ovenproof skillet. Add roulades and cook until browned on all sides, 8-10 minutes. Transfer skillet to oven and bake until an instant-read thermometer inserted into center of roulades registers 165°, 7-8 minutes. (The chicken will be cooked through but still juicy.) Transfer chicken to plates and let rest for 10 minutes.

Scrape drippings and any melted cheese from skillet; discard. Set skillet over medium-high heat and melt 1 tablespoon butter. Add mushrooms. Cook, turning once, until mushrooms are golden brown, about 5 minutes. Season with salt and pepper; add broth and vinegar. Simmer until liquid is thickened and reduced, 10-12 minutes. Stir in remaining 1 tablespoon butter and herbs; season with salt and pepper.

Cut off and discard twine. Cut chicken into ½-inch slices. Drizzle mushroom sauce over.