Turn on broiler. Combine cucumber and 1/3 cup butter in a bowl. Set aside. Combine remaining butter with parsley and half the lemon juice. Set aside. Arrange flounder in a single layer in an oiled shallow broiling pan. Sprinkle with remaining lemon juice and white pepper. Smear cucumber butter evenly over fillets. Broil 3 inches from heat source 5-8 minutes, basting once, until fish flakes easily. Place a spoonful of parsley butter over hot fish and serve sprinkled with cayenne.
Per serving: calories 321, fat 21.9g, 62% calories from fat, cholesterol 122mg, protein 27.4g, carbohydrates 3.3g, fiber 1.0g, sodium 120mg.
The recommended wines are: Riesling, Chablis, or Gewürztraminer.