Sprinkle each breast with sage and pepper. Place 1 slice of prosciutto on each chicken piece.
Heat 1 tablespoon butter in large, heavy nonstick skillet over high heat. When butter starts to brown, add chicken, prosciutto side down; sauté 2 minutes per side or until no longer pink in center. Transfer to platter. Cover; keep warm.
Add vermouth and broth to skillet; cook, scraping up browned bits from bottom, 2 minutes. Add remaining tablespoon butter; swirl until just incorporated into a creamy sauce. Pour over chicken.