Prep the chicken: If using whole chicken breasts, slice each horizontally into two thinner cutlets, then pound between two sheets of plastic wrap to an even 1/4-inch thickness. Pat the cutlets dry with paper towels and season both sides with salt and pepper.
Dredge in flour: Spread the flour in a shallow dish. Press each cutlet firmly into the flour, coating both sides, then shake off the excess. Set the dredged cutlets on a plate while the pan heats.
Sear until golden: Heat the olive oil with 2 tablespoons of butter in a large skillet over medium-high heat until the butter foams. Add the cutlets in a single layer and sear for 2.5-3 minutes per side until deep golden and cooked through. Transfer to a warm plate and tent loosely with foil.
Deglaze and build the sauce: Pour off excess oil from the skillet, leaving the browned bits. Return the pan to medium heat and pour in the white wine, scraping up every flavorful bit with a wooden spoon. Let it bubble for 2 minutes to reduce slightly. Add the chicken broth, lemon juice, and capers; simmer for 3-4 minutes until the sauce reduces by about a third.
Finish with cold butter: Reduce the heat to low. Add the cold butter pieces one at a time, swirling the pan constantly until each melts and the sauce turns silky and glossy. Stir in the parsley and taste: add a pinch more salt or a squeeze of lemon if needed.
Serve: Return the cutlets (and any resting juices) to the pan, spooning the lemon caper sauce over each piece. Simmer for just 1 minute to warm through, then serve immediately with lemon slices and extra sauce drizzled on top.