Place eggs in 4-quart saucepan. Add enough cold water to cover eggs by 2 inches. Heat to boiling on high, cover with lid, and remove from heat. Let stand 12 minutes. Drain and rinse under cold water. Under running water, peel off shells and discard. Trim the tops and bottoms of eggs so that they can sit flat. Cut each egg in half crosswise. Transfer yolks to large bowl. Into yolks, mash mayonnaise, cayenne, and 1/4 teaspoon salt until smooth. Stir in pepper, carrot, cornichons, and dill until well mixed. Spoon rounded teaspoon of yolk mixture into each egg-white half. Place on serving platter; garnish with dill. Eggs can be made up to 1 day ahead, then covered and refrigerated.