1. Place flour in a shallow dish. Place milk in another shallow dish. Sprinkle fish evenly with 1/4 teaspoon salt and 1/4 teaspoon pepper. Working with one fillet at a time, dredge fish in flour, shaking off excess. Dip fish into milk, allowing excess to drip off; dredge again in flour. Set aside. Repeat procedure with remaining fish, flour, and milk.
2. Heat a large nonstick skillet over medium-high heat. Add 1 tablespoon butter to pan, swirling until butter melts. Add 2 fillets to pan; reduce heat to medium, and cook 4 minutes on each side or until golden brown and fish flakes easily when tested with a fork or until desired degree of doneness. Remove fish from pan; keep warm. Repeat procedure with remaining 1 tablespoon butter and the remaining 2 fillets. Remove fish from pan; keep warm.
3. Increase heat to medium-high. Add shallots and remaining 1/4 teaspoon salt to pan; sauté 1 minute or until tender. Add vinegar and broth; simmer 1 minute. Add parsley and remaining 1/4 teaspoon pepper. Spoon sauce over fish.
Wine note: This polished preparation with its savory butter flavors needs a light, crisp, unoaked white wine that will act as the "lemon" to balance the flavors of the fish. Patient Cottat Sauvignon Blanc 2006 Vin de Pays du Jardin de la France is a standout and a steal at $13. —Karen MacNeil