Heat the oven to 450°F. In a small bowl, combine the Parmesan, prosciutto, sun-dried tomatoes, sage, and 1 Tbs. of the butter and mix until well blended. Taste and season with pepper and possibly more salt, though the prosciutto and cheese contribute salt. The mixture should be highly seasoned.
Gently slide your index finger under the skin to create a small pocket. Scoop up some of the filling and push it into the pocket, spreading it as evenly as possible. Do this in a few places until you've used half the filling and covered the top of the breast, but be careful not to completely detach the skin from the meat. Tidy up the skin by stretching it over the filling and the breast as evenly as possible. Repeat with the second breast. Chill in the freezer for about 5 minutes to let the filling firm up.
Remove the chicken from the freezer, and season both sides with salt and pepper. Heat a cast-iron or other heavy-based ovenproof skillet on the stove over medium-high heat. When the pan is hot, add the oil and the last 1 tsp. butter. When the butter stops foaming, add the chicken breasts, skin side down. Don't try to move them for at least 1 minute or the skin might tear. After 1 minute or so, you can move them around to be sure they're not sticking. Cook until the skin side is well browned, about 4 minutes.
Carefully slide a thin spatula under the chicken and flip it over, taking care not to rip the skin. Put the pan in the hot oven and continue cooking until the chicken is no longer pink inside, about 10 minutes or until a thermometer reads 165°F at the thickest part. Take the breasts from the oven, let them rest for 3 to 5 minutes tented with foil, and then serve immediately.