Soak fillets in milk for 20 minutes. Remove from milk and pat dry.
Season fillets with salt and pepper.
Heat 1 tablespoon oil over moderate heat in each of 2 large saute pans until hot.
Put flour into a shallow dish and season with salt and pepper. Dredge fillets in flour, shaking off any excess flour. Dip the fillets back into the milk and then again into the flour. Place fillets in saute pans and cook over medium heat for 2 to 3 minutes per side, until golden brown and crispy. Transfer fillets from pans to a platter, sprinkle with parsley.
Wipe the skillet clean and add the butter. Heat until the butter begins to brown. Remove from heat and stir in balsamic vinegar. Pour sauce over fish and serve. Garnish with lemon slices.