Cut each chicken breast in half crosswise to make 4 portions. Cut a slit into the sides of each portion. Insert a slice of prosciutto, a slice of cheese and 2 whole basil leaves into each slit. Secure well with a toothpick.
Heat the oil in a large non-stick skillet over medium-high heat until hot; add the chicken. Reduce heat to medium; cook, turning once, until chicken is browned, about 3 minutes per side.
Transfer chicken to a baking pan.
Bake until chicken is cooked through, about 15-18 minutes.
Transfer to a platter.
Meanwhile, heat the broth, mustard and garlic to a boil in the same skillet over high heat; cook until slightly thickened, 5 minutes. Stir in sliced basil; pour over chicken.