Pound the chicken breasts between sheets of plastic wrap to an even 1/2-inch thickness using a meat mallet or rolling pin.

Mix the flour, salt, and pepper in a shallow bowl. Dredge each chicken breast lightly, coating both sides and shaking off any excess.

Heat the olive oil and 2 tablespoons butter in a large skillet over medium-high heat until the butter foams. Add the chicken and cook for 3 to 4 minutes per side until deep golden brown and cooked through. Remove to a plate and tent with foil.

Pour the chicken stock or wine and lemon juice into the hot skillet. Scrape the bottom thoroughly with a wooden spoon to dissolve the fond. Add the capers and bring to a boil. Simmer for 3 to 5 minutes until reduced by about half.

Reduce heat to very low. Add the cold butter cubes one at a time, whisking continuously until each cube fully incorporates before adding the next. The sauce should become glossy and slightly thickened. Never let the pan return to a boil during this step.

Stir in the fresh parsley. Return the chicken to the pan for 1 minute on low heat, spooning the sauce over each cutlet to glaze.

Plate immediately with extra sauce spooned over and fresh lemon slices alongside.