Place fillets in a shallow pan; pour milk over fillets. Cover and refrigerate 10 minutes, turning once. Combine flour, salt, and pepper, stirring well. Drain fillets; dredge in flour mixture.
Heat 1/4 inch oil in a large skillet. Add fillets; cook 3 minutes on each side or until browned, turning once. Transfer fillets to a serving platter; set aside, and keep warm.
Melt butter in a small skillet over low heat; cook, stirring frequently, until butter is browned. Remove from heat, and stir in lemon juice. Immediately pour butter mixture over fillets; sprinkle with parsley, and garnish with cucumber slices. Serve immediately.