Stir together flour, salt, and pepper in a shallow dish. Place beaten egg white in a separate shallow dish. Dip each flounder fillet in egg white, and dredge in flour mixture, shaking off any excess.
Heat 2 tablespoons of the butter and 1 tablespoon of the oil in a large nonstick skillet over medium-high until butter melts. Add 2 fillets, and cook until golden brown, 2 to 3 minutes per side. Transfer to a plate. Wipe skillet clean, and repeat process with 2 tablespoons of the butter and remaining 1 tablespoon olive oil and 2 fillets. Discard drippings; do not wipe skillet clean.
Add broth, lemon juice, and capers to skillet. Bring to a boil over high, stirring and scraping to loosen browned bits from bottom of skillet. Reduce heat to medium, and simmer until reduced to about 1/3 cup, about 5 minutes. Whisk in parsley and remaining 4 tablespoons butter.
Place flounder fillets on a platter. Spoon sauce over fish, and serve immediately with Italian Salsa Verde.