Place chicken breasts between 2 sheets of heavy plastic on a solid, level surface. Firmly pound chicken breasts with the smooth side of a meat mallet to a 1/2-inch thickness.

Season both sides of chicken breasts with cayenne, salt, and black pepper; dredge lightly in flour and shake off any excess.

Heat olive oil in a skillet over medium-high heat. Place chicken in the pan, reduce heat to medium, and cook until browned and cooked through, about 5 minutes per side; transfer to a plate.

Cook capers in reserved oil, smashing them lightly to release brine, until warmed through, about 30 seconds. Pour white wine into the skillet and bring to a boil while scraping the browned bits of food off the bottom of the pan with a wooden spoon. Cook until reduced by half, about 2 minutes.

Stir lemon juice, water, and butter into the reduced wine mixture; cook and stir continuously to form a thick sauce, about 2 minutes. Reduce heat to low and stir parsley through the sauce.

Return chicken breasts to the pan and cook until heated through, 1 to 2 minutes. Serve with sauce spooned over the top.