How to Make It
Place each chicken cutlet between two sheets of heavy-duty plastic wrap, and flatten to 1/4-inch thickness, using a rolling pin or flat side of a meat mallet. Stir together flour, salt, and pepper. Dip each cutlet in egg white, and dredge in flour mixture, shaking off excess.
Melt 2 tablespoons butter with 1 tablespoon olive oil in a large cast-iron skillet over medium-high. Add 2 cutlets, and cook until golden brown, 2 to 3 minutes on each side. Transfer to a plate. Wipe skillet clean, and repeat process with 2 tablespoons butter and remaining 1 tablespoon olive oil and remaining 2 cutlets. Discard drippings; do not wipe skillet clean.
Add broth, lemon juice, and capers to skillet. Bring to a boil over high, stirring and scraping bottom of skillet to loosen browned bits. Reduce heat to medium, and simmer, whisking occasionally, 5 minutes. Whisk in remaining 2 tablespoons butter. Whisk in parsley. Spoon sauce over chicken, and serve immediately with pasta.