Combine all ingredients in a small bowl and stir well to combine. Set aside. Sprinkle the breasts with the rub, reserving one teaspoon.
Lay two prosciutto slices on a work surface, overlapping so that the width of the two is slightly smaller than the width of your chicken breasts. Place a chicken breast (with the smooth side down) on top of the prosciutto and wrap it around so that it covers most of the chicken.
Repeat with the remaining prosciutto and chicken. Lay the prosciutto-wrapped breasts with the smooth side up on the work surface. Sprinkle each lightly with the remaining rub. Allow breasts to sit with the rub on for 15-20 minutes, at room temperature.
Heat the butter and olive oil in a cast iron or heavy stainless steel pan over medium heat. Butter should sizzle a bit and be fairly hot, but watch it at this point to prevent burning. Make sure the pan is large enough to accommodate all of the breasts without crowding.
Add the chicken (smooth side down) and allow to sit for 3 minutes without touching. After 3 minutes, take a peek at the underside of one of the breasts. If it’s browning too quickly lower the heat a bit. Continue cooking for a total of 5 minutes or until the underside is nicely browned.
Flip breasts to the opposite side (seam side down) and immediately lower heat to medium-low. Cover and continue to cook for another 5-10 minutes, until internal temperature reaches 160˚F when an instant thermometer is inserted into the center of the thickest part of the breast. (The time will vary depending on the size of each portion - very large breasts will take longer. Use the thermometer as your guide, not the time.)
Once 160˚F is reached, remove the chicken to a clean plate and cover it loosely with foil. You can stop at this point and let rest for 5 minutes before serving OR proceed with the sauce, letting the chicken rest while it's being made.
Add ½ cup wine to the pan and bring to a boil, stirring and scraping any brown particles (fond) from the bottom of the pan. Cook until the wine is reduced by one half, 1-2 minutes.
Add the chicken broth, return to a boil and continue cooking for 1-2 minutes until the sauce starts to thicken a bit. Add the cold butter one piece at a time, stirring until each piece is melted before adding the next. Stir in one teaspoon of finely chopped fresh rosemary.
Transfer the chicken to a platter and (optionally) drizzle with the sauce. Sprinkle with more finely chopped fresh rosemary and garnish with rosemary sprigs.
See Café Tips above in the post for more detailed instructions and tips.