Begin by removing the small tenderloin from each chicken breast, that small muscle on the underside of the breast. Slice chicken breast horizontally, so that you wind up with two or three slices per breast depending on thickness.
Take each slice, place it between two sheets of parchment paper, then pound it gently with a meat pounder or the back of a skillet. Do the same with the tenderloin. The resulting scalloppine should be about 1/2 cm or 1/4 inch.
Lightly flour the scalloppine. Heat a skillet with olive oil and a knob of butter over moderate heat. When the butter has melted and stopped bubbling, add the scalloppine and brown them lightly on both sides. Season them with salt on both sides, giving them a final turn as you do. Remove and keep warm.
Pour the white wine into the skillet and scrap up the bottom of the skillet. Let the wine simmer for a minute or two, then add the freshly squeezed lemon juice and capers.
Return the scalloppine to the pan and turn them in the sauce for just a few moments, until they're heated through and the sauce has thickened a bit. Transfer the scalloppine to a serving dish or platter.
Turn off the flame. Swirl in a dab of butter and the minced parsley into the sauce until it reaches a creamy consistence. Pour over the scalloppine and top with some additional minced parsley if you like.