Peel and slice the peaches. Set aside.
Pound the pork medallions with the side of a mallet until about ¼ to ½-inch in thickness.
Sprinkle salt and pepper over both sides of the pieces of pork.
Dredge each seasoned piece of pork through a shallow bowl filled with the flour. Shake off excess flour and place on a platter.
In a large skillet, preferably cast-iron, heat the oil over medium heat. Once the oil is shimmering, add the cutlets to the pan and cook until browned on the bottom, about 3 to 4 minutes. Carefully shuffle the cutlets with tongs or a spatula a little to help them from sticking to the pan.
Carefully flip the cutlets and cook for another 3 to 4 minutes, until browned all over and cooked through. Remove the cutlets to a platter or large plate.
With the pan still over medium heat, add the sherry to the pan. Stir with a wooden spoon, and prying up any bits that may be stuck to the pan from searing the pork cutlets. Cook, stirring frequently, until reduced, about 1 minute.
Add the scallions and capers to the pan and stir for about 1 minute. Add the chicken stock and then peaches. Cook until the sauce has slightly thickened, about 8 to 10 minutes. Season with ½ tsp salt and a pinch of pepper.
Nestle the pork cutlets into the sauce and cook for another couple of minutes, enough to warm the cutlets. Stir in the lemon juice and then serve at once.