1. Place eggs in a medium saucepan and cover with cold water. Bring water to a boil and immediately remove from heat. Cover and let eggs stand in hot water for 10 to 12 minutes. Remove from hot water, cool and peel.
2. Slice eggs in half lengthwise. Remove yolks and place in a medium bowl. Set aside egg whites. Mashing with a fork, mix mayonnaise, Dijon-style mustard, salt and black pepper with the egg yolks.
3. Fill the hollowed egg white halves with the yolk mixture. Sprinkle with cayenne pepper, adjusting the amount to taste. Cover and chill in the refrigerator until serving.