Flatten the chicken breasts so they are relatively the same thickness, about ¼ inch thick. You can use a meat hammer or pounder if you have one. If not, try a rolling pin or the side of a can of soup.
Place the chicken breasts between wax paper or plastic wrap before pounding.
Season flour with salt and pepper.
Dredge the chicken breasts into the flour and shake off any extra flour so it doesn't clump in the pan. Be sure the chicken breasts are dry before dredging. You need to do this just before cooking. You can't prep this ahead of time.
Get your fry pan / skillet hot over medium-high heat, add 2 tablespoons of butter and 2 tablespoons of oil.
When the oil and butter are hot but not smoking, add the chicken breasts and pan fry for 2-3 minutes on one side, then flip and cook on the other side for 2-3 minutes.
If the pan isn't large enough to cook all four at one time, cook them in two stages or use two pans.
Remove and reserve chicken to a platter and cover with tin foil to keep warm.
Deglaze the pan with wine, loosening all of the browned bits stuck to the pan with a wooden spoon.
Add the lemon juice, chicken stock, capers and bring to a boil. Lower the heat to medium and reduce the sauce by half.
Add the remaining 2 tablespoons of butter and whisk to incorporate.
Taste and adjust seasoning with salt and pepper.
Add half the chopped parsley, reserving the rest for plating.
Plate the chicken breasts along with your sides, sauce and serve immediately.