Preheat oven to 300°F. Combine cheeses, prosciutto, and basil in small bowl. Season stuffing with freshly ground black pepper.

Run fingers under skin of chicken breasts to loosen. Spread stuffing under skin. Sprinkle chicken with salt and pepper.

Melt 1 tablespoon butter in large nonstick skillet over medium-high heat. Add chicken, skin side down, and cook until brown, about 3 minutes per side. Reduce heat to medium, cover, and cook until chicken is cooked through, turning once, about 10 minutes. Transfer to rimmed baking sheet; place in oven to keep warm.

Add broth and Marsala to skillet; bring to boil and cook until sauce is reduced to scant ½ cup scraping up browned bits, 3 to 4 minutes. Remove from heat; whisk in 1 tablespoon butter. Season sauce to taste with salt and pepper. Place chicken on plates. Add any juices from baking sheet to sauce. Spoon sauce over chicken and serve.