Kosher salt & freshly ground black pepper to taste
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. Spread 1 tablespoon of Mascarpone on chicken. Place 2 slices of prosciutto on top, leaving a ½-inch border. Top prosciutto with 3 slices of Provolone. Spread 1 tablespoon of pesto on top, maintaining ½-inch border. Roll up chicken lengthwise and tie with kitchen twine. Repeat with remaining chicken breasts.
Preheat oven to 450 degrees F. Heat 1 tablespoon of 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 degrees F, 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 and discard. Set skillet over medium-high heat and melt 1 tablespoon of butter. Add mushrooms. Cook, turning once, until mushrooms are golden brown, about 5 minutes. Season with salt and pepper, then add broth and vinegar. Simmer until liquid is thickened and reduced, 10 – 12 minutes. Stir in ¼ cup of heavy cream, and herbs. Season with salt and pepper.
Cut off and discard twine. Cut chicken into ½-inch slices. Drizzle mushroom sauce over chicken slices.