Gently run your fingers between the skin and meat of the chicken breasts, taking care not to tear the skin as you loosen it. Tuck the rosemary-prosciutto mixture under the skin of the chicken breasts. Rub the skin all over with 2 tablespoons softened butter; season with salt and pepper.
Place chicken breasts, skin side up, in flameproof roasting pan large enough to hold them in one layer. Roast for 15 minutes. Reduce heat to 375 degrees; add garlic cloves to pan and roast until internal temperature of chicken is 170 degrees on instant-read thermometer, about 25 minutes more. Spoon the accumulated juices over the chicken once halfway through this cooking time.
Transfer chicken to a platter, cover with foil, and set aside. Add chicken broth and sherry to roasting pan and place pan over medium-high heat. Bring to boil, stirring to loosen any browned bits from the bottom of pan while mashing roasted garlic with back of spoon. Boil until liquid is reduced by half, 3 to 4 minutes. Stir in remaining butter to thicken sauce. Arrange chicken on plates, top with sauce and serve.