Squeeze the juice out of one of the lemons and set the juice aside. From the other lemon, slice four very thin slices from the center (the widest part), and set those aside, too.
Stretch a piece of plastic wrap across your work surface, place the chicken breast filets on it, and lay another piece of plastic wrap on top. Then, using a meat mallet, rolling pin or similar object, flatten the chicken breasts between the layers of plastic wrap until they are about ¼ inch thick.
In a shallow baking dish or even a plate, combine the flour, salt, and pepper.
Heat a heavy-bottomed sauté pan over medium-high heat for a minute, then add the olive oil and heat for another 30 seconds or so. Add the onion and garlic to the pan and sauté until the onion is slightly translucent, about 3 minutes. Remove the onion and garlic from the pan and set them aside.
Add the butter to the pan and let it heat until it turns foamy.
dredge both sides of the chicken breast filets in the flour mixture, shake off any excess flour and add them, one at a time, to the hot pan. Cook 2 to 3 minutes per side or until they are nicely browned. Remove them from the pan and set aside on a plate, covered with foil, while you make the sauce.
Add the stock or broth, sherry, lemon juice and capers to the pan and scrape off any flavorful bits from the bottom of the pan. Then return the onion-garlic mixture to the pan and heat it through, adding a bit more oil or butter if necessary.
Return the chicken to the pan, bring the liquid to a boil, then lower it to a simmer and cook for about 3 minutes or until the mixture has reduced by about one-third. Add the chopped parsley just at the end of the cooking.
With a pair of tongs, remove the chicken breasts and plate them. Adjust the seasoning on the sauce with Kosher salt and lemon juice. Sauce generously, top each portion with a lemon slice and serve immediately.
Serve with orzo pasta or white rice sprinkled with parsley, salt, and pepper, and halved grape tomatoes.