Prepare the grill for direct cooking over medium heat (350° to 450°F).
Using a sharp knife, butterfly each pork chop by making a slit halfway up the fat side of the chop and enlarging the slit to extend to within ½ inch of the opposite side. Do not cut all the way through. Open each chop like a book and season the inside with salt and pepper.
Lay one slice of prosciutto inside each chop, folding the prosciutto to fit, if necessary. Close the chops and press tightly. Lightly coat the outside of the chops with oil and season evenly with salt and pepper.
Brush the cooking grates clean. Grill the pork chops over direct medium heat, with the lid closed, until nicely browned on the outside with just a trace of pink on the inside, about 8 minutes, turning once or twice. Meanwhile, make the sauce.
In a small skillet over medium heat, melt the butter and cook until bubbly, 2 to 3 minutes. Add the sage and cook for about 1 minute. Remove the skillet from the heat. Allow the butter to cool for about 1 minute. Add the lemon juice to the melted butter, and then immediately spoon the sauce over the chops. Serve right away.